Mathletic | Practice Platform

AMC 12·2000 · AMC 12A·Easy

In the year

2001

, the United States will host the International Mathematical Olympiad. Let

I,M,

and

O

be distinct positive integers such that the product

I \cdot M \cdot O = 2001

. What is the largest possible value of the sum

I + M + O

?

AMC 12·2000 · AMC 12A·Medium

Two non-zero real numbers,

a

and

b,

satisfy

ab = a - b

. Which of the following is a possible value of

\frac {a}{b} + \frac {b}{a} - ab

?

AMC 12·2000 · AMC 12A·Medium

Let

A, M,

and

C

be nonnegative integers such that

A + M + C = 12

. What is the maximum value of

A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C

?

AMC 12·2000 · AMC 12A·Easy

2000(2000^{2000}) =

AMC 12·2000 · AMC 12A·Hard

The graph below shows a portion of the curve defined by the quartic polynomial

P(x) = x^4 + ax^3 + bx^2 + cx + d

. Which of the following is the smallest?

AMC 12·2000 · AMC 12A·Easy

Each day, Jenny ate

20\%

of the jellybeans that were in her jar at the beginning of that day. At the end of the second day,

32

remained. How many jellybeans were in the jar originally?

AMC 12·2000 · AMC 12A·Easy

The Fibonacci sequence

1,1,2,3,5,8,13,21,\ldots

starts with two

1

's, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

AMC 12·2000 · AMC 12A·Easy

If

|x - 2| = p,

where

x < 2,

then

x - p =

AMC 12·2000 · AMC 12A·Easy

Figures

0

,

1

,

2

, and

3

consist of

1

,

5

,

13

, and

25

nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?

AMC 12·2000 · AMC 12A·Easy

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were

71,76,80,82,

and

91

. What was the last score Mrs. Walters entered?

AMC 12·2001 · AMC 12A·Easy

The sum of two numbers is

S

. Suppose

3

is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

AMC 12·2001 · AMC 12A·Easy

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

AMC 12·2001 · AMC 12A·Medium

The parabola with equation

y=ax^2+bx+c

and vertex

(h,k)

is reflected about the line

y=k

. This results in the parabola with equation

y=dx^2+ex+f

. Which of the following equals

a+b+c+d+e+f

?

AMC 12·2001 · AMC 12A·Medium

The polynomial

P(x)=x^3+ax^2+bx+c

has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the

y

-intercept of the graph of

y=P(x)

is 2, what is

b

?

AMC 12·2001 · AMC 12A·Hard

A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?

AMC 12·2001 · AMC 12A·Easy

The state income tax where Kristin lives is levied at the rate of

p\%

of the first

\$28000

of annual income plus

(p + 2)\%

of any amount above

\$28000

. Kristin noticed that the state income tax she paid amounted to

(p + 0.25)\%

of her annual income. What was her annual income?

AMC 12·2001 · AMC 12A·Easy

The mean of three numbers is

10

more than the least of the numbers and

15

less than the greatest. The median of the three numbers is

5

. What is their sum?

AMC 12·2001 · AMC 12A·Easy

What is the product of all positive odd integers less than

10000

?

AMC 12·2001 · AMC 12A·Easy

A telephone number has the form

\text{ABC-DEF-GHIJ}

, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is,

A > B > C

,

D > E > F

, and

G > H > I > J

. Furthermore,

D

,

E

, and

F

are consecutive even digits;

G

,

H

,

I

, and

J

are consecutive odd digits; and

A + B + C = 9

. Find

A

.

AMC 12·2001 · AMC 12A·Easy

A charity sells

140

benefit tickets for a total of

\$2001

. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?

AMC 12·2001 · AMC 12A·Easy

Let

f

be a function satisfying

f(xy) = \frac{f(x)}y

for all positive real numbers

x

and

y

. If

f(500) =3

, what is the value of

f(600)

?

AMC 12·2002 · AMC 12A·Easy

Compute the sum of all the roots of

(2x+3)(x-4)+(2x+3)(x-6)=0

AMC 12·2002 · AMC 12A·Easy

Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

AMC 12·2002 · AMC 12A·Medium

Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?

AMC 12·2002 · AMC 12A·Medium

Both roots of the quadratic equation

x^2 - 63x + k = 0

are prime numbers. The number of possible values of

k

is

AMC 12·2002 · AMC 12A·Medium

Two different positive numbers

a

and

b

each differ from their reciprocals by

1

. What is

a+b

?

AMC 12·2002 · AMC 12A·Medium

For all positive integers

n

, let

f(n)=\log_{2002} n^2

. Let

N=f(11)+f(13)+f(14)

. Which of the following relations is true?

AMC 12·2002 · AMC 12A·Medium

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

AMC 12·2002 · AMC 12A·Medium

The graph of the function

f

is shown below. How many solutions does the equation

f(f(x))=6

have?

AMC 12·2002 · AMC 12A·Easy

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?

AMC 12·2002 · AMC 12A·Hard

Find the number of ordered pairs of real numbers

(a,b)

such that

(a+bi)^{2002} = a-bi

.

AMC 12·2002 · AMC 12A·Easy

According to the standard convention for exponentiation, 2^{2^{2^{2}}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65536. If the order in which the exponentiations are performed is changed, how many other values are possible?

AMC 12·2002 · AMC 12A·Easy

Find the degree measure of an angle whose complement is 25% of its supplement.

AMC 12·2002 · AMC 12A·Easy

Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?

AMC 12·2002 · AMC 12B·Easy

The arithmetic mean of the nine numbers in the set

\{9, 99, 999, 9999, \ldots, 999999999\}

is a

9

-digit number

M

, all of whose digits are distinct. The number

M

does not contain the digit

AMC 12·2002 · AMC 12B·Medium

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

AMC 12·2002 · AMC 12B·Medium

How many four-digit numbers

N

have the property that the three-digit number obtained by removing the leftmost digit is one ninth of

N

?

AMC 12·2002 · AMC 12B·Medium

Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?

AMC 12·2002 · AMC 12B·Medium

If

a,b,

and

c

are positive real numbers such that

a(b+c) = 152, b(c+a) = 162,

and

c(a+b) = 170

, then

abc

is

AMC 12·2002 · AMC 12B·Easy

What is the value of (3x - 2)(4x + 1) - (3x - 2)4x + 1 when

x=4

?

AMC 12·2002 · AMC 12B·Hard

For all integers

n

greater than

1

, define

a_n = \frac{1}{\log_n 2002}

. Let

b = a_2 + a_3 + a_4 + a_5

and

c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}

. Then

b- c

equals

AMC 12·2002 · AMC 12B·Hard

Let

f(x) = x^2 + 6x + 1

, and let

R

denote the set of points

(x,y)

in the coordinate plane such that f(x) + f(y) \le 0 \text{and} f(x)-f(y) \le 0 The area of

R

is closest to

AMC 12·2002 · AMC 12B·Easy

Let

n

be a positive integer such that

\frac 12 + \frac 13 + \frac 17 + \frac 1n

is an integer. Which of the following statements is '''not''' true:

AMC 12·2002 · AMC 12B·Easy

Suppose that

a

and

b

are nonzero real numbers, and that the equation

x^2 + ax + b = 0

has solutions

a

and

b

. Then the pair

(a,b)

is

AMC 12·2002 · AMC 12B·Easy

The product of three consecutive positive integers is

8

times their sum. What is the sum of their squares?

AMC 12·2002 · AMC 12B·Easy

Suppose July of year

N

has five Mondays. Which of the following must occur five times in August of year

N

? (Note: Both months have 31 days.)

AMC 12·2002 · AMC 12B·Easy

If

a,b,c,d

are positive real numbers such that

a,b,c,d

form an increasing arithmetic sequence and

a,b,d

form a geometric sequence, then

\frac ad

is

AMC 12·2002 · AMC 12 Practice·Easy

Let

f_n (x) = \text{\sin}^n x + \text{\cos}^n x.

For how many

x

in

[0,\pi]

is it true that

6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?

AMC 12·2002 · AMC 12 Practice·Medium

Let

t_n = \frac{n(n+1)}{2}

be the

n

th triangular number. Find

\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}

AMC 12·2002 · AMC 12 Practice·Hard

Let

a

and

b

be real numbers such that

\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}

and

\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.

Find

\sin{(a+b)}.

AMC 12·2003 · AMC 12A·Easy

What is the difference between the sum of the first

2003

even counting numbers and the sum of the first

2003

odd counting numbers?

AMC 12·2003 · AMC 12A·Easy

Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of

3:2:1

, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?

AMC 12·2003 · AMC 12A·Medium

A parabola with equation

y=ax^2+bx+c

is reflected about the

x

-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of

y=f(x)

and

y=g(x)

, respectively. Which of the following describes the graph of

y=(f+g)(x)?

AMC 12·2003 · AMC 12A·Easy

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?

AMC 12·2003 · AMC 12A·Hard

The graph of the polynomial

P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e

has five distinct

x

-intercepts, one of which is at

(0,0)

. Which of the following coefficients cannot be zero?

AMC 12·2003 · AMC 12A·Hard

If

a\geq b > 1,

what is the largest possible value of

\log_{a}(a/b) + \log_{b}(b/a)?

AMC 12·2003 · AMC 12A·Easy

It takes Mary

30

minutes to walk uphill

1

km from her home to school, but it takes her only

10

minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?

AMC 12·2003 · AMC 12A·Easy

The sum of the two 5-digit numbers

AMC10

and

AMC12

is

123422

. What is

A+M+C

?

AMC 12·2003 · AMC 12A·Easy

Define

x \heartsuit y

to be

|x-y|

for all real numbers

x

and

y

. Which of the following statements is not true?

AMC 12·2003 · AMC 12A·Easy

A set

S

of points in the

xy

-plane is symmetric about the origin, both coordinate axes, and the line

y=x

. If

(2,3)

is in

S

, what is the smallest number of points in

S

?

AMC 12·2003 · AMC 12B·Medium

If

\log (xy^3) = 1

and

\log (x^2y) = 1

, what is

\log (xy)

?

AMC 12·2003 · AMC 12B·Medium

Part of the graph of

f(x) = ax^3 + bx^2 + cx + d

is shown. What is

b

?

AMC 12·2003 · AMC 12B·Hard

The number of

x

-intercepts on the graph of

y=\sin(1/x)

in the interval

(0.0001,0.001)

is closest to

AMC 12·2003 · AMC 12B·Hard

Positive integers

a,b,

and

c

are chosen so that

a<b<c

, and the system of equations <center>

2x + y = 2003

and

y = |x-a| + |x-b| + |x-c|

</center> has exactly one solution. What is the minimum value of

c

?

AMC 12·2004 · AMC 12A·Easy

Alicia earns

20

dollars per hour, of which

1.45\%

is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?

AMC 12·2004 · AMC 12A·Easy

On the AMC 12, each correct answer is worth

6

points, each incorrect answer is worth

0

points, and each problem left unanswered is worth

2.5

points. If Charlyn leaves

8

of the

25

problems unanswered, how many of the remaining problems must she answer correctly in order to score at least

100

?

AMC 12·2004 · AMC 12A·Easy

Bertha has

6

daughters and no sons. Some of her daughters have

6

daughters, and the rest have none. Bertha has a total of

30

daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?

AMC 12·2004 · AMC 12A·Easy

The graph of the line

y=mx+b

is shown. Which of the following is true?

AMC 12·2004 · AMC 12A·Easy

Let

U=2\cdot 2004^{2005}

,

V=2004^{2005}

,

W=2003\cdot 2004^{2004}

,

X=2\cdot 2004^{2004}

,

Y=2004^{2004}

and

Z=2004^{2003}

. Which of the following is the largest?

AMC 12·2004 · AMC 12A·Easy

A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players

A

,

B

and

C

start with

15

,

14

and

13

tokens, respectively. How many rounds will there be in the game?

AMC 12·2004 · AMC 12B·Medium

All the students in an algebra class took a

100

-point test. Five students scored

100

, each student scored at least

60

, and the mean score was

76

. What is the smallest possible number of students in the class?

AMC 12·2004 · AMC 12B·Medium

In the sequence

2001

,

2002

,

2003

,

\ldots

, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is

2001 + 2002 - 2003 = 2000

. What is the

2004^\textrm{th}

term in this sequence?

AMC 12·2004 · AMC 12B·Medium

If

f(x) = ax+b

and

f^{-1}(x) = bx+a

with

a

and

b

real, what is the value of

a+b

?

AMC 12·2004 · AMC 12B·Medium

The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?

AMC 12·2004 · AMC 12B·Medium

A function

f

is defined by

f(z) = i\overline{z}

, where

i=\sqrt{-1}

and

\overline{z}

is the complex conjugate of

z

. How many values of

z

satisfy both

|z| = 5

and

f(z) = z

?

AMC 12·2004 · AMC 12B·Medium

For some real numbers

a

and

b

, the equation 8x^3 + 4ax^2 + 2bx + a = 0 has three distinct positive roots. If the sum of the base-

2

logarithms of the roots is

5

, what is the value of

a

?

AMC 12·2004 · AMC 12B·Medium

Points

A

and

B

are on the parabola

y=4x^2+7x-1

, and the origin is the midpoint of

AB

. What is the length of

AB

?

AMC 12·2004 · AMC 12B·Hard

The graph of

2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0

is an ellipse in the first quadrant of the

xy

-plane. Let

a

and

b

be the maximum and minimum values of

\frac yx

over all points

(x,y)

on the ellipse. What is the value of

a+b

?

AMC 12·2004 · AMC 12B·Hard

The square <center>

\begin{array}{c} \hline 50 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 2 \\ \hline \end{array}

</center> is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of

g

?

AMC 12·2004 · AMC 12B·Hard

The polynomial

x^3 - 2004 x^2 + mx + n

has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of

n

are possible?

AMC 12·2005 · AMC 12A·Easy

Two is

10 \%

of

x

and

20 \%

of

y

. What is

x - y

?

AMC 12·2005 · AMC 12A·Medium

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

AMC 12·2005 · AMC 12A·Medium

A line passes through

A(1,1)

and

B(100,1000)

. How many other points with integer coordinates are on the line and strictly between

A

and

B

?

AMC 12·2005 · AMC 12A·Medium

In the five-sided star shown, the letters

A

,

B

,

C

,

D

and

E

are replaced by the numbers

3

,

5

,

6

,

7

and

9

, although not necessarily in that order. The sums of the numbers at the ends of the line segments

\overline{AB}

,

\overline{BC}

,

\overline{CD}

,

\overline{DE}

and

\overline{EA}

form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

AMC 12·2005 · AMC 12A·Medium

A faulty car odometer proceeds from digit

3

to digit

5

, always skipping the digit

4

, regardless of position. For example, after traveling one mile the odometer changed from

000039

to

000050

. If the odometer now reads

002005

, how many miles has the car actually traveled?

AMC 12·2005 · AMC 12A·Easy

The equations

2x + 7 = 3

and

bx - 10 = - 2

have the same solution for

x

. What is the value of

b

?

AMC 12·2005 · AMC 12A·Medium

For each

x

in

[0,1]

, define f(x) = \begin{cases} 2x, \mathrm{if} 0 \leq x \leq \frac{1}{2}\\ 2-2x, \mathrm{if} \frac{1}{2} < x \leq 1. \end{cases} Let

f^{[2]}(x) = f(f(x))

, and

f^{[n + 1]}(x) = f^{[n]}(f(x))

for each integer

n \geq 2

. For how many values of

x

in

[0,1]

is

f^{[2005]}(x) = 1/2

?

AMC 12·2005 · AMC 12A·Hard

How many ordered triples of integers

(a,b,c)

, with

a \geq 2

,

b \geq 1

, and

c \geq 0

, satisfy both

\log_{a}b = c^{2005}

and

a + b + c = 2005

?

AMC 12·2005 · AMC 12A·Hard

Let

P(x) = (x - 1)(x - 2)(x - 3)

. For how many polynomials

Q(x)

does there exist a polynomial

R(x)

of degree

3

such that

P(Q(x)) = P(x) \cdot R(x)

?

AMC 12·2005 · AMC 12A·Easy

A store normally sells windows at

\$100

each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?

AMC 12·2005 · AMC 12A·Easy

The average (mean) of

20

numbers is

30

, and the average of

30

other numbers is

20

. What is the average of all

50

numbers?

AMC 12·2005 · AMC 12A·Easy

Josh and Mike live

13

miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

AMC 12·2005 · AMC 12A·Easy

Let

A

,

M

, and

C

be digits with (100A+10M+C)(A+M+C) = 2005. What is

A

?

AMC 12·2005 · AMC 12A·Easy

There are two values of

a

for which the equation

4x^2 + ax + 8x + 9 = 0

has only one solution for

x

. What is the sum of those values of

a

?

AMC 12·2005 · AMC 12B·Easy

A scout troop buys

1000

candy bars at a price of five for

2

dollars. They sell all the candy bars at the price of two for

1

dollar. What was their profit, in dollars?

AMC 12·2005 · AMC 12B·Medium

The quadratic equation

x^2+mx+n

has roots twice those of

x^2+px+m

, and none of

m,n,

and

p

is zero. What is the value of

n/p

?

AMC 12·2005 · AMC 12B·Medium

Suppose that

4^{x_1}=5

,

5^{x_2}=6

,

6^{x_3}=7

, ... ,

127^{x_{124}}=128

. What is

x_1x_2...x_{124}

?

AMC 12·2005 · AMC 12B·Medium

How many distinct four-tuples

(a, b, c, d)

of rational numbers are there with

a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005

?

AMC 12·2005 · AMC 12B·Easy

A positive number

x

has the property that

x\%

of

x

is

4

. What is

x

?

AMC 12·2005 · AMC 12B·Medium

Let

a,b,c,d,e,f,g

and

h

be distinct elements in the set

\{-7,-5,-3,-2,2,4,6,13\}.

What is the minimum possible value of

(a+b+c+d)^{2}+(e+f+g+h)^{2}?

AMC 12·2005 · AMC 12B·Hard

Let

S

be the set of ordered triples

(x,y,z)

of real numbers for which

\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.

There are real numbers

a

and

b

such that for all ordered triples

(x,y,z)

in

S

we have

x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.

What is the value of

a+b?

AMC 12·2005 · AMC 12B·Easy

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

AMC 12·2005 · AMC 12B·Easy

At the beginning of the school year, Lisa's goal was to earn an A on at least

80\%

of her

50

quizzes for the year. She earned an A on

22

of the first

30

quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

AMC 12·2005 · AMC 12B·Easy

For how many values of

a

is it true that the line

y = x + a

passes through the vertex of the parabola

y = x^2 + a^2

?

AMC 12·2005 · AMC 12B·Easy

On a certain math exam,

10\%

of the students got

70

points,

25\%

got

80

points,

20\%

got

85

points,

15\%

got

90

points, and the rest got

95

points. What is the difference between the mean and the median score on this exam?

AMC 12·2006 · AMC 12A·Easy

Sandwiches at Joe's Fast Food cost

3

dollars each and sodas cost

2

dollars each. How many dollars will it cost to purchase

5

sandwiches and

8

sodas?

AMC 12·2006 · AMC 12A·Easy

For how many real values of

x

is

\sqrt{120-\sqrt{x}}

an integer?

AMC 12·2006 · AMC 12A·Medium

Which of the following describes the graph of the equation

(x+y)^2=x^2+y^2

?

AMC 12·2006 · AMC 12A·Medium

Two farmers agree that pigs are worth

300

dollars and that goats are worth

210

dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a

390

dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

AMC 12·2006 · AMC 12A·Medium

Suppose

\cos x=0

and

\cos (x+z)=1/2

. What is the smallest possible positive value of

z

?

AMC 12·2006 · AMC 12A·Medium

The function

f

has the property that for each real number

x

in its domain,

1/x

is also in its domain and

f(x)+f\left(\frac{1}{x}\right)=x

What is the largest set of real numbers that can be in the domain of

f

?

AMC 12·2006 · AMC 12A·Easy

Define

x\otimes y=x^3-y

. What is

h\otimes (h\otimes h)

?

AMC 12·2006 · AMC 12A·Hard

Let

S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}

and

S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}

. What is the ratio of the area of

S_2

to the area of

S_1

?

AMC 12·2006 · AMC 12A·Hard

Given a finite sequence

S=(a_1,a_2,\ldots ,a_n)

of

n

real numbers, let

A(S)

be the sequence

\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)

of

n-1

real numbers. Define

A^1(S)=A(S)

and, for each integer

m

,

2\le m\le n-1

, define

A^m(S)=A(A^{m-1}(S))

. Suppose

x>0

, and let

S=(1,x,x^2,\ldots ,x^{100})

. If

A^{100}(S)=(1/2^{50})

, then what is

x

?

AMC 12·2006 · AMC 12A·Hard

The expression

(x+y+z)^{2006}+(x-y-z)^{2006}

is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

AMC 12·2006 · AMC 12A·Easy

The ratio of Mary's age to Alice's age is

3:5

. Alice is

30

years old. How old is Mary?

AMC 12·2006 · AMC 12A·Easy

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?

AMC 12·2006 · AMC 12A·Easy

Doug and Dave shared a pizza with

8

equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was

8

dollars, and there was an additional cost of

2

dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?

AMC 12·2006 · AMC 12A·Easy

Mary is

20\%

older than Sally, and Sally is

40\%

younger than Danielle. The sum of their ages is

23.2

years. How old will Mary be on her next birthday?

AMC 12·2006 · AMC 12A·Easy

How many sets of two or more consecutive positive integers have a sum of

15

?

AMC 12·2006 · AMC 12A·Easy

Oscar buys

13

pencils and

3

erasers for

\$1.00

. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?

AMC 12·2006 · AMC 12B·Medium

Elmo makes

N

sandwiches for a fundraiser. For each sandwich he uses

B

globs of peanut butter at

4

cents per glob and

J

blobs of jam at

5

cents per blob. The cost of the peanut butter and jam to make all the sandwiches is

\$2.53

. Assume that

B

,

J

and

N

are all positive integers with

N>1

. What is the cost of the jam Elmo uses to make the sandwiches?

AMC 12·2007 · AMC 12A·Easy

One ticket to a show costs

\$20

at full price. Susan buys

4

tickets using a coupon that gives her a

25\%

discount. Pam buys

5

tickets using a coupon that gives her a

30\%

discount. How many more dollars does Pam pay than Susan?

AMC 12·2007 · AMC 12A·Medium

A piece of cheese is located at

(12,10)

in a coordinate plane. A mouse is at

(4,-2)

and is running up the line

y=-5x+18

. At the point

(a,b)

the mouse starts getting farther from the cheese rather than closer to it. What is

a+b

?

AMC 12·2007 · AMC 12A·Medium

Let a, b, c, d, and e be distinct integers such that <center>

(6-a)(6-b)(6-c)(6-d)(6-e)=45

</center> What is

a+b+c+d+e

?

AMC 12·2007 · AMC 12A·Medium

The set

\{3,6,9,10\}

is augmented by a fifth element

n

, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of

n

?

AMC 12·2007 · AMC 12A·Medium

How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?

AMC 12·2007 · AMC 12A·Medium

Suppose that

\sin a + \sin b = \sqrt{\frac{5}{3}}

and

\cos a + \cos b = 1

. What is

\cos (a - b)

?

AMC 12·2007 · AMC 12A·Medium

The polynomial

f(x) = x^{4} + ax^{3} + bx^{2} + cx + d

has real coefficients, and

f(2i) = f(2 + i) = 0.

What is

a + b + c + d?

AMC 12·2007 · AMC 12A·Medium

Triangles

ABC

and

ADE

have areas

2007

and

7002,

respectively, with

B = (0,0), C = (223,0), D = (680,380),

and

E = (689,389).

What is the sum of all possible x-coordinates of

A

?

AMC 12·2007 · AMC 12A·Easy

An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?

AMC 12·2007 · AMC 12A·Hard

Square

ABCD

has area

36,

and

\overline{AB}

is parallel to the x-axis. Vertices

A, B

, and

C

are on the graphs of

y = \log_{a}x, y = 2\log_{a}x,

and

y = 3\log_{a}x,

respectively. What is

a?

AMC 12·2007 · AMC 12A·Hard

For each integer

n>1

, let

F(n)

be the number of solutions to the equation

\sin{x}=\sin{(nx)}

on the interval

[0,\pi]

. What is

\sum_{n=2}^{2007} F(n)

?

AMC 12·2007 · AMC 12A·Hard

Call a set of integers ''spacy'' if it contains no more than one out of any three consecutive integers. How many subsets of

\{1,2,3,\ldots,12\},

including the empty set, are spacy?

AMC 12·2007 · AMC 12A·Easy

The larger of two consecutive odd integers is three times the smaller. What is their sum?

AMC 12·2007 · AMC 12A·Easy

Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?

AMC 12·2007 · AMC 12A·Easy

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?

AMC 12·2007 · AMC 12B·Easy

Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?

AMC 12·2008 · AMC 12A·Easy

A bakery owner turns on his doughnut machine at

\text{8:30}\ {\small\text{AM}}

. At

\text{11:10}\ {\small\text{AM}}

the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?

AMC 12·2008 · AMC 12A·Easy

Doug can paint a room in

5

hours. Dave can paint the same room in

7

hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let

t

be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by

t

?

AMC 12·2008 · AMC 12A·Medium

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the

13

visible numbers have the greatest possible sum. What is that sum?

AMC 12·2008 · AMC 12A·Medium

A function

f

has domain

[0,2]

and range

[0,1]

. (The notation

[a,b]

denotes

\{x:a \le x \le b \}

.) What are the domain and range, respectively, of the function

g

defined by

g(x)=1-f(x+1)

?

AMC 12·2008 · AMC 12A·Medium

The numbers

\log(a^3b^7)

,

\log(a^5b^{12})

, and

\log(a^8b^{15})

are the first three terms of an arithmetic sequence , and the

12^\text{th}

term of the sequence is

\log{b^n}

. What is

n

?

AMC 12·2008 · AMC 12A·Medium

In the expansion of

\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,

what is the coefficient of

x^{28}

?

AMC 12·2008 · AMC 12A·Easy

What is the reciprocal of

\frac{1}{2}+\frac{2}{3}

?

AMC 12·2008 · AMC 12A·Hard

A sequence

(a_1,b_1)

,

(a_2,b_2)

,

(a_3,b_3)

,

\ldots

of points in the coordinate plane satisfies

(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)

for

n = 1,2,3,\ldots

. Suppose that

(a_{100},b_{100}) = (2,4)

. What is

a_1 + b_1

?

AMC 12·2008 · AMC 12A·Easy

Suppose that

\tfrac{2}{3}

of

10

bananas are worth as much as

8

oranges. How many oranges are worth as much as

\tfrac{1}{2}

of

5

bananas?

AMC 12·2008 · AMC 12A·Easy

Which of the following is equal to the product \frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?

AMC 12·2008 · AMC 12A·Easy

Suppose that \frac{2x}{3}-\frac{x}{6} is an integer. Which of the following statements must be true about

x

?

AMC 12·2008 · AMC 12A·Easy

Heather compares the price of a new computer at two different stores. Store

A

offers

15\%

off the sticker price followed by a

\$90

rebate, and store

B

offers

25\%

off the same sticker price with no rebate. Heather saves

\$15

by buying the computer at store

A

instead of store

B

. What is the sticker price of the computer, in dollars?

AMC 12·2008 · AMC 12A·Easy

While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?

AMC 12·2008 · AMC 12A·Easy

Older television screens have an aspect ratio of

4: 3

. That is, the ratio of the width to the height is

4: 3

. The aspect ratio of many movies is not

4: 3

, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of

2: 1

and is shown on an older television screen with a

27

-inch diagonal. What is the height, in inches, of each darkened strip?

AMC 12·2008 · AMC 12B·Easy

Bricklayer Brenda would take

9

hours to build a chimney alone, and bricklayer Brandon would take

10

hours to build it alone. When they work together they talk a lot, and their combined output is decreased by

10

bricks per hour. Working together, they build the chimney in

5

hours. How many bricks are in the chimney? (Solution)

AMC 12·2008 · AMC 12B·Medium

For each positive integer

n

, the mean of the first

n

terms of a sequence is

n

. What is the

2008

th term of the sequence?

AMC 12·2008 · AMC 12B·Medium

A function

f

is defined by

f(z) = (4 + i) z^2 + \alpha z + \gamma

for all complex numbers

z

, where

\alpha

and

\gamma

are complex numbers and

i^2 = - 1

. Suppose that

f(1)

and

f(i)

are both real. What is the smallest possible value of

| \alpha | + |\gamma |

? (Solution)

AMC 12·2008 · AMC 12B·Easy

A

4\times 4

block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?

\begin{array}{c} \multicolumn{4}{c}{}\\\hline 1&2&3&4\\\hline 8&9&10&11\\\hline 15&16&17&18\\\hline 22&23&24&25\\\hline \end{array}

AMC 12·2008 · AMC 12B·Medium

Michael walks at the rate of

5

feet per second on a long straight path. Trash pails are located every

200

feet along the path. A garbage truck travels at

10

feet per second in the same direction as Michael and stops for

30

seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? (Solution)

AMC 12·2008 · AMC 12B·Easy

A semipro baseball league has teams with

21

players each. League rules state that a player must be paid at least

15,000

dollars, and that the total of all players' salaries for each team cannot exceed

700,000

dollars. What is the maximum possible salary, in dollars, for a single player?

AMC 12·2008 · AMC 12B·Easy

Postman Pete has a pedometer to count his steps. The pedometer records up to

99999

steps, then flips over to

00000

on the next step. Pete plans to determine his mileage for a year. On January

1

Pete sets the pedometer to

00000

. During the year, the pedometer flips from

99999

to

00000

forty-four times. On December

31

the pedometer reads

50000

. Pete takes

1800

steps per mile. Which of the following is closest to the number of miles Pete walked during the year?

AMC 12·2008 · AMC 12B·Easy

For real numbers

a

and

b

, define

a

\ b = (a - b)^2

. What is

(x - y)^2

\ (y - x)^2

? (Solution)

AMC 12·2008 · AMC 12B·Easy

Points

B

and

C

lie on

\overline{AD}

. The length of

\overline{AB}

is

4

times the length of

\overline{BD}

, and the length of

\overline{AC}

is

9

times the length of

\overline{CD}

. The length of

\overline{BC}

is what fraction of the length of

\overline{AD}

? (Solution)

AMC 12·2009 · AMC 12A·Easy

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took

h

hours and

m

minutes, with

0 < m < 60

, what is

h + m

?

AMC 12·2009 · AMC 12A·Medium

The figures

F_1

,

F_2

,

F_3

, and

F_4

shown are the first in a sequence of figures. For

n\ge3

,

F_n

is constructed from

F_{n - 1}

by surrounding it with a square and placing one more diamond on each side of the new square than

F_{n - 1}

had on each side of its outside square. For example, figure

F_3

has

13

diamonds. How many diamonds are there in figure

F_{20}

?

AMC 12·2009 · AMC 12A·Medium

A ship sails

10

miles in a straight line from

A

to

B

, turns through an angle between

45^{\circ}

and

60^{\circ}

, and then sails another

20

miles to

C

. Let

AC

be measured in miles. Which of the following intervals contains

AC^2

?

AMC 12·2009 · AMC 12A·Medium

For what value of

n

is

i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i

? Note: here

i = \sqrt { - 1}

.

AMC 12·2009 · AMC 12A·Medium

Let

a + ar_1 + ar_1^2 + ar_1^3 + \cdots

and

a + ar_2 + ar_2^2 + ar_2^3 + \cdots

be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is

r_1

, and the sum of the second series is

r_2

. What is

r_1 + r_2

?

AMC 12·2009 · AMC 12A·Easy

Which of the following is equal to

1 + \frac {1}{1 + \frac {1}{1 + 1}}

?

AMC 12·2009 · AMC 12A·Hard

Let

p(x) = x^3 + ax^2 + bx + c

, where

a

,

b

, and

c

are complex numbers. Suppose that <center>

p(2009 + 9002\pi i) = p(2009) = p(9002) = 0

</center> What is the number of nonreal zeros of

x^{12} + ax^8 + bx^4 + c

?

AMC 12·2009 · AMC 12A·Hard

Functions

f

and

g

are quadratic,

g(x) = - f(100 - x)

, and the graph of

g

contains the vertex of the graph of

f

. The four

x

-intercepts on the two graphs have

x

-coordinates

x_1

,

x_2

,

x_3

, and

x_4

, in increasing order, and

x_3 - x_2 = 150

. Then

x_4 - x_1 = m + n\sqrt p

, where

m

,

n

, and

p

are positive integers, and

p

is not divisible by the square of any prime. What is

m + n + p

?

AMC 12·2009 · AMC 12A·Hard

The ''tower function of twos'' is defined recursively as follows:

T(1) = 2

and

T(n + 1) = 2^{T(n)}

for

n\ge1

. Let

A = (T(2009))^{T(2009)}

and

B = (T(2009))^A

. What is the largest integer

k

such that \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ \times}} is defined?

AMC 12·2009 · AMC 12A·Hard

The first two terms of a sequence are

a_1 = 1

and

a_2 = \frac {1}{\sqrt3}

. For

n\ge1

, <center> a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}. </center> What is

|a_{2009}|

?

AMC 12·2009 · AMC 12A·Easy

What number is one third of the way from

\frac14

to

\frac34

?

AMC 12·2009 · AMC 12A·Easy

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?

AMC 12·2009 · AMC 12A·Easy

Suppose that

P = 2^m

and

Q = 3^n

. Which of the following is equal to

12^{mn}

for every pair of integers

(m,n)

?

AMC 12·2009 · AMC 12A·Easy

The first three terms of an arithmetic sequence are

2x - 3

,

5x - 11

, and

3x + 1

respectively. The

n

th term of the sequence is

2009

. What is

n

?

AMC 12·2009 · AMC 12A·Easy

Suppose that

f(x+3)=3x^2 + 7x + 4

and

f(x)=ax^2 + bx + c

. What is

a+b+c

?

AMC 12·2009 · AMC 12B·Easy

Each morning of her five-day workweek, Jane bought either a

50

-cent muffin or a

75

-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?

AMC 12·2009 · AMC 12B·Easy

A particular

12

-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a

1

, it mistakenly displays a

9

. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?

AMC 12·2009 · AMC 12B·Medium

On Monday, Millie puts a quart of seeds,

25\%

of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only

25\%

of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?

AMC 12·2009 · AMC 12B·Medium

The fifth and eighth terms of a geometric sequence of real numbers are

7!

and

8!

respectively. What is the first term?

AMC 12·2009 · AMC 12B·Medium

Assume

0 < r < 3

. Below are five equations for

x

. Which equation has the largest solution

x

?

AMC 12·2009 · AMC 12B·Easy

Paula the painter had just enough paint for

30

identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for

25

rooms. How many cans of paint did she use for the

25

rooms?

AMC 12·2009 · AMC 12B·Hard

For how many values of

x

in

[0,\pi]

is

\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)

? Note: The functions

\sin^{ - 1} = \arcsin

and

\cos^{ - 1} = \arccos

denote inverse trigonometric functions.

AMC 12·2009 · AMC 12B·Easy

Twenty percent less than

60

is one-third more than what number?

AMC 12·2009 · AMC 12B·Easy

Kiana has two older twin brothers. The product of their ages is

128

. What is the sum of their three ages?

AMC 12·2009 · AMC 12B·Easy

In a certain year the price of gasoline rose by

20\%

during January, fell by

20\%

during February, rose by

25\%

during March, and fell by

x\%

during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is

x

?

AMC 12·2009 · AMC 12B·Easy

When a bucket is two-thirds full of water, the bucket and water weigh

a

kilograms. When the bucket is one-half full of water the total weight is

b

kilograms. In terms of

a

and

b

, what is the total weight in kilograms when the bucket is full of water?

AMC 12·2010 · AMC 12A·Easy

What is

\left(20-\left(2010-201\right)\right)+\left(2010-\left(201-20\right)\right)

?

AMC 12·2010 · AMC 12A·Easy

The first four terms of an arithmetic sequence are

p

,

9

,

3p-q

, and

3p+q

. What is the

2010^\text{th}

term of this sequence?

AMC 12·2010 · AMC 12A·Medium

The solution of the equation

7^{x+7} = 8^x

can be expressed in the form

x = \log_b 7^7

. What is

b

?

AMC 12·2010 · AMC 12A·Medium

In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements. Brian: "Mike and I are different species." Chris: "LeRoy is a frog." LeRoy: "Chris is a frog." Mike: "Of the four of us, at least two are toads." How many of these amphibians are frogs?

AMC 12·2010 · AMC 12A·Medium

For how many integer values of

k

do the graphs of

x^2+y^2=k^2

and

xy = k

not intersect?

AMC 12·2010 · AMC 12A·Easy

A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?

AMC 12·2010 · AMC 12A·Hard

The graph of

y=x^6-10x^5+29x^4-4x^3+ax^2

lies above the line

y=bx+c

except at three values of

x

, where the graph and the line intersect. What is the largest of these values?

AMC 12·2010 · AMC 12A·Hard

What is the minimum value of

f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|

?

AMC 12·2010 · AMC 12A·Hard

Let

f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)

. The intersection of the domain of

f(x)

with the interval

[0,1]

is a union of

n

disjoint open intervals. What is

n

?

AMC 12·2010 · AMC 12A·Easy

If

x<0

, then which of the following must be positive?

AMC 12·2010 · AMC 12A·Easy

Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next

n

shots are bullseyes she will be guaranteed victory. What is the minimum value for

n

?

AMC 12·2010 · AMC 12B·Easy

Makarla attended two meetings during her

9

-hour work day. The first meeting took

45

minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?

AMC 12·2010 · AMC 12B·Easy

The average of the numbers

1, 2, 3,\cdots, 98, 99,

and

x

is

100x

. What is

x

?

AMC 12·2010 · AMC 12B·Medium

For what value of

x

does

\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?

AMC 12·2010 · AMC 12B·Medium

Let

a

,

b

,

c

,

d

, and

e

be positive integers with

a+b+c+d+e=2010

and let

M

be the largest of the sum

a+b

,

b+c

,

c+d

and

d+e

. What is the smallest possible value of

M

?

AMC 12·2010 · AMC 12B·Medium

The entries in a

3 \times 3

array include all the digits from

1

through

9

, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

AMC 12·2010 · AMC 12B·Medium

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than

100

points. What was the total number of points scored by the two teams in the first half?

AMC 12·2010 · AMC 12B·Easy

A big

L

is formed as shown. What is its area?

AMC 12·2010 · AMC 12B·Medium

A geometric sequence

(a_n)

has

a_1=\sin x

,

a_2=\cos x

, and

a_3= \tan x

for some real number

x

. For what value of

n

does

a_n=1+\cos x

?

AMC 12·2010 · AMC 12B·Hard

Let

a > 0

, and let

P(x)

be a polynomial with integer coefficients such that <center>

P(1) = P(3) = P(5) = P(7) = a

, and

P(2) = P(4) = P(6) = P(8) = -a

. </center> What is the smallest possible value of

a

?

AMC 12·2010 · AMC 12B·Hard

Monic quadratic polynomial

P(x)

and

Q(x)

have the property that

P(Q(x))

has zeros at

x=-23, -21, -17,

and

-15

, and

Q(P(x))

has zeros at

x=-59,-57,-51

and

-49

. What is the sum of the minimum values of

P(x)

and

Q(x)

?

AMC 12·2010 · AMC 12B·Hard

The set of real numbers

x

for which

\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1

is the union of intervals of the form

a<x\le b

. What is the sum of the lengths of these intervals?

AMC 12·2010 · AMC 12B·Easy

A ticket to a school play cost

x

dollars, where

x

is a whole number. A group of 9th graders buys tickets costing a total of $

48

, and a group of 10th graders buys tickets costing a total of $

64

. How many values for

x

are possible?

AMC 12·2010 · AMC 12B·Easy

A month with

31

days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

AMC 12·2010 · AMC 12B·Easy

Lucky Larry's teacher asked him to substitute numbers for

a

,

b

,

c

,

d

, and

e

in the expression

a-(b-(c-(d+e)))

and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for

a

,

b

,

c

, and

d

were

1

,

2

,

3

, and

4

, respectively. What number did Larry substitute for

e

?

AMC 12·2010 · AMC 12B·Easy

At the beginning of the school year,

50\%

of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and

50\%

answered "No." At the end of the school year,

70\%

answered "Yes" and

30\%

answered "No." Altogether,

x\%

of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of

x

?

AMC 12·2010 · AMC 12B·Easy

Shelby drives her scooter at a speed of

30

miles per hour if it is not raining, and

20

miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of

16

miles in

40

minutes. How many minutes did she drive in the rain?

AMC 12·2011 · AMC 12A·Easy

A cell phone plan costs

\$20

dollars each month, plus

5

cents per \text message sent, plus

10

cents for each minute used over

30

hours. In January Michelle sent

100

\text messages and talked for

30.5

hours. How much did she have to pay?

AMC 12·2011 · AMC 12A·Medium

A power boat and a raft both left dock

A

on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock

B

downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock

A.

How many hours did it take the power boat to go from

A

to

B

?

AMC 12·2011 · AMC 12A·Medium

Suppose that

\left|x+y\right|+\left|x-y\right|=2

. What is the maximum possible value of

x^2-6x+y^2

?

AMC 12·2011 · AMC 12A·Medium

At a competition with

N

players, the number of players given elite status is equal to

2^{1+\lfloor \log_{2} (N-1) \rfloor}-N

. Suppose that

19

players are given elite status. What is the sum of the two smallest possible values of

N

?

AMC 12·2011 · AMC 12A·Medium

Let

f(x)=ax^2+bx+c

, where

a

,

b

, and

c

are integers. Suppose that

f(1)=0

,

50<f(7)<60

,

70<f(8)<80

,

5000k<f(100)<5000(k+1)

for some integer

k

. What is

k

?

AMC 12·2011 · AMC 12A·Hard

Let

f_{1}(x)=\sqrt{1-x}

, and for integers

n \geq 2

, let

f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})

. If

N

is the largest value of

n

for which the domain of

f_{n}

is nonempty, the domain of

f_{N}

is

\{c\}

. What is

N+c

?

AMC 12·2011 · AMC 12A·Hard

Let

f(z)= \frac{z+a}{z+b}

and

g(z)=f(f(z))

, where

a

and

b

are complex numbers. Suppose that

\left| a \right| = 1

and

g(g(z))=z

for all

z

for which

g(g(z))

is defined. What is the difference between the largest and smallest possible values of

\left| b \right|

?

AMC 12·2011 · AMC 12A·Easy

A small bottle of shampoo can hold

35

milliliters of shampoo, whereas a large bottle can hold

500

milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?

AMC 12·2011 · AMC 12A·Easy

At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of

12

,

15

, and

10

minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?

AMC 12·2011 · AMC 12A·Easy

Last summer

30\%

of the birds living on Town Lake were geese,

25\%

were swans,

10\%

were herons, and

35\%

were ducks. What percent of the birds that were not swans were geese?

AMC 12·2011 · AMC 12A·Easy

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was

61

points. How many free throws did they make?

AMC 12·2011 · AMC 12A·Easy

A majority of the

30

students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than

1

. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was

17.71

. What was the cost of a pencil in cents?

AMC 12·2011 · AMC 12A·Easy

In the eight term sequence

A

,

B

,

C

,

D

,

E

,

F

,

G

,

H

, the value of

C

is

5

and the sum of any three consecutive terms is

30

. What is

A+H

?

AMC 12·2011 · AMC 12A·Easy

At a twins and triplets convention, there were

9

sets of twins and

6

sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?

AMC 12·2011 · AMC 12B·Easy

What is

\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?

AMC 12·2011 · AMC 12B·Medium

Brian writes down four integers

w > x > y > z

whose sum is

44

. The pairwise positive differences of these numbers are

1, 3, 4, 5, 6

and

9

. What is the sum of the possible values of

w

?

AMC 12·2011 · AMC 12B·Medium

Let

f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))

, and

h_n(x) = h_1(h_{n-1}(x))

for integers

n \geq 2

. What is the sum of the digits of

h_{2011}(1)

?

AMC 12·2011 · AMC 12B·Easy

Josanna's test scores to date are

90, 80, 70, 60,

and

85.

Her goal is to raise her test average at least

3

points with her next test. What is the minimum test score she would need to accomplish this goal?

AMC 12·2011 · AMC 12B·Hard

A bug travels in the coordinate plane, moving only along the lines that are parallel to the

x

-axis or

y

-axis. Let

A = (-3, 2)

and

B = (3, -2)

. Consider all possible paths of the bug from

A

to

B

of length at most

20

. How many points with integer coordinates lie on at least one of these paths?

AMC 12·2011 · AMC 12B·Hard

Let

P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)

. What is the minimum perimeter among all the

8

-sided polygons in the complex plane whose vertices are precisely the zeros of

P(z)

?

AMC 12·2011 · AMC 12B·Easy

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where

A < B.

How many dollars must LeRoy give to Bernardo so that they share the costs equally?

AMC 12·2011 · AMC 12B·Easy

Let

x

and

y

be two-digit positive integers with mean

60

. What is the maximum value of the ratio

\frac{x}{y}

?

AMC 12·2012 · AMC 12A·Easy

A bug crawls along a number line, starting at

-2

. It crawls to

-6

, then turns around and crawls to

5

. How many units does the bug crawl altogether?

AMC 12·2012 · AMC 12A·Medium

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at

\text{8:00 AM}

, and all three always take the same amount of time to eat lunch. On Monday the three of them painted

50\%

of a house, quitting at

\text{4:00 PM}

. On Tuesday, when Paula wasn't there, the two helpers painted only

24\%

of the house and quit at

\text{2:12 PM}

. On Wednesday Paula worked by herself and finished the house by working until

\text{7:12 PM}

. How long, in minutes, was each day's lunch break?

AMC 12·2012 · AMC 12A·Easy

Cagney can frost a cupcake every

20

seconds and Lacey can frost a cupcake every

30

seconds. Working together, how many cupcakes can they frost in

5

minutes?

AMC 12·2012 · AMC 12A·Medium

Consider the polynomial

P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)

The coefficient of

x^{2012}

is equal to

2^a

. What is

a

? \[

AMC 12·2012 · AMC 12A·Hard

Let

a

,

b

, and

c

be positive integers with

a\ge b\ge c

such that

\begin{align*}a^{2}-b^{2}-c^{2}+ab&=2011\text{ and}\\ a^{2}+3b^{2}+3c^{2}-3ab-2ac-2bc&=-1997 \end{align*}

What is

a

?

AMC 12·2012 · AMC 12A·Hard

Let

\{a_k\}_{k=1}^{2011}

be the sequence of real numbers defined by

a_1=0.201, a_2=(0.2011)^{a_1}, a_3=(0.20101)^{a_2}, a_4=(0.201011)^{a_3}

, and in general,

a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}} \text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}} \text{if }k\text{ is even.}\end{cases}

Rearranging the numbers in the sequence

\{a_k\}_{k=1}^{2011}

in decreasing order produces a new sequence

\{b_k\}_{k=1}^{2011}

. What is the sum of all integers

k

,

1\le k \le 2011

, such that

a_k=b_k?

AMC 12·2012 · AMC 12A·Hard

Let

f(x)=|2\{x\}-1|

where

\{x\}

denotes the fractional part of

x

. The number

n

is the smallest positive integer such that the equation

nf(xf(x))=x

has at least

2012

real solutions. What is

n

? Note: the fractional part of

x

is a real number

y=\{x\}

such that

0\le y<1

and

x-y

is an integer.

AMC 12·2012 · AMC 12A·Easy

A box

2

centimeters high,

3

centimeters wide, and

5

centimeters long can hold

40

grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold

n

grams of clay. What is

n

?

AMC 12·2012 · AMC 12A·Easy

In a bag of marbles,

\tfrac{3}{5}

of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?

AMC 12·2012 · AMC 12A·Easy

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of

280

pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?

AMC 12·2012 · AMC 12A·Easy

The sums of three whole numbers taken in pairs are

12

,

17

, and

19

. What is the middle number?

AMC 12·2012 · AMC 12A·Easy

Mary divides a circle into

12

sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

AMC 12·2012 · AMC 12A·Easy

An ''iterative average'' of the numbers

1

,

2

,

3

,

4

, and

5

is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?

AMC 12·2012 · AMC 12B·Easy

Each third-grade classroom at Pearl Creek Elementary has

18

students and

2

pet rabbits. How many more students than rabbits are there in all

4

of the third-grade classrooms?

AMC 12·2012 · AMC 12B·Medium

In the equation below,

A

and

B

are consecutive positive integers, and

A

,

B

, and

A+B

represent number bases:

132_A+43_B=69_{A+B}.

What is

A+B

?

AMC 12·2012 · AMC 12B·Medium

Let

(a_1,a_2, \dots ,a_{10})

be a list of the first

10

positive integers such that for each

2 \le i \le 10

either

a_i+1

or

a_i-1

or both appear somewhere before

a_i

in the list. How many such lists are there?

AMC 12·2012 · AMC 12B·Hard

Consider all polynomials of a complex variable,

P(z)=4z^4+az^3+bz^2+cz+d

, where

a,b,c,

and

d

are integers,

0\le d\le c\le b\le a\le 4

, and the polynomial has a zero

z_0

with

|z_0|=1.

What is the sum of all values

P(1)

over all the polynomials with these properties?

AMC 12·2012 · AMC 12B·Easy

For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?

AMC 12·2012 · AMC 12B·Easy

Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?

AMC 12·2012 · AMC 12B·Easy

Two integers have a sum of

26

. when two more integers are added to the first two, the sum is

41

. Finally, when two more integers are added to the sum of the previous

4

integers, the sum is

57

. What is the minimum number of even integers among the

6

integers?

AMC 12·2012 · AMC 12B·Easy

In order to estimate the value of

x-y

where

x

and

y

are real numbers with

x>y>0

, Xiaoli rounded

x

up by a small amount, rounded

y

down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

AMC 12·2012 · AMC 12B·Easy

Small lights are hung on a string

6

inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of

2

red lights followed by

3

green lights. How many feet separate the 3rd red light and the 21st red light? Note:

1

foot is equal to

12

inches.

AMC 12·2012 · AMC 12B·Easy

It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking?

AMC 12·2013 · AMC 12A·Easy

Let

S

be the set of positive integers

n

for which

\tfrac{1}{n}

has the repeating decimal representation

0.\overline{ab} = 0.ababab\cdots,

with

a

and

b

different digits. What is the sum of the elements of

S

?

AMC 12·2013 · AMC 12A·Medium

The sequence

\log_{12}{162}

,

\log_{12}{x}

,

\log_{12}{y}

,

\log_{12}{z}

,

\log_{12}{1250}

is an arithmetic progression. What is

x

?

AMC 12·2013 · AMC 12A·Medium

A

,

B

,

C

are three piles of rocks. The mean weight of the rocks in

A

is

40

pounds, the mean weight of the rocks in

B

is

50

pounds, the mean weight of the rocks in the combined piles

A

and

B

is

43

pounds, and the mean weight of the rocks in the combined piles

A

and

C

is

44

pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles

B

and

C

?

AMC 12·2013 · AMC 12A·Hard

Consider

A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots ))))

. Which of the following intervals contains

A

?

AMC 12·2013 · AMC 12A·Hard

Let

f : \mathbb{C} \to \mathbb{C}

be defined by

f(z) = z^2 + iz + 1

. How many complex numbers

z

are there such that

\text{Im}(z) > 0

and both the real and the imaginary parts of

f(z)

are integers with absolute value at most

10

?

AMC 12·2013 · AMC 12A·Easy

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

AMC 12·2013 · AMC 12A·Easy

What is the value of

\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?

AMC 12·2013 · AMC 12A·Easy

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid 105

, Dorothy paid 125

, and Sammy paid 175

. In order \to share the costs equally, Tom gave Sammy

t

dollars, and Dorothy gave Sammy

d

dollars. What is

t-d

?

AMC 12·2013 · AMC 12A·Easy

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on

20\%

of her three-point shots and

30\%

of her two-point shots. Shenille attempted

30

shots. How many points did she score?

AMC 12·2013 · AMC 12A·Easy

The sequence

S_1, S_2, S_3, \cdots, S_{10}

has the property that every term beginning with the third is the sum of the previous two. That is,

S_n = S_{n-2} + S_{n-1} \text{ for } n \ge 3.

Suppose that

S_9 = 110

and

S_7 = 42

. What is

S_4

?

AMC 12·2013 · AMC 12A·Easy

Given that

x

and

y

are distinct nonzero real numbers such that

x+\tfrac{2}{x} = y + \tfrac{2}{y}

, what is

xy

?

AMC 12·2013 · AMC 12B·Easy

On a particular January day, the high temperature in Lincoln, Nebraska, was

16

degrees higher than the low temperature, and the average of the high and low temperatures was

3

. In degrees, what was the low temperature in Lincoln that day?

AMC 12·2013 · AMC 12B·Easy

Alex has

75

red tokens and

75

blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

AMC 12·2013 · AMC 12B·Medium

Two bees start at the same spot and fly at the same rate in the following directions. Bee

A

travels

1

foot north, then

1

foot east, then

1

foot upwards, and then continues to repeat this pattern. Bee

B

travels

1

foot south, then

1

foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly

10

feet away from each other?

AMC 12·2013 · AMC 12B·Medium

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is

N

. What is the smallest possible value of

N

?

AMC 12·2013 · AMC 12B·Medium

The number

2013

is expressed in the form <center>

2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}

,</center> where

a_1 \ge a_2 \ge ... \ge a_m

and

b_1 \ge b_2 \ge ... \ge b_n

are positive integers and

a_1 + b_1

is as small as possible. What is

|a_1 - b_1|

?

AMC 12·2013 · AMC 12B·Medium

Let

a,b,

and

c

be real numbers such that

a+b+c=2, \text{ and}

a^2+b^2+c^2=12

What is the difference between the maximum and minimum possible values of

c

?

AMC 12·2013 · AMC 12B·Easy

Mr. Green measures his rectangular garden by walking two of the sides and finds that it is

15

steps by

20

steps. Each of Mr. Green’s steps is

2

feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?

AMC 12·2013 · AMC 12B·Hard

Let

m>1

and

n>1

be integers. Suppose that the product of the solutions for

x

of the equation

8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0

is the smallest possible integer. What is

m+n

?

AMC 12·2013 · AMC 12B·Hard

Let

G

be the set of polynomials of the form

P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,

where

c_1,c_2,\cdots, c_{n-1}

are integers and

P(z)

has distinct roots of the form

a+ib

with

a

and

b

integers. How many polynomials are in

G

?

AMC 12·2013 · AMC 12B·Easy

When counting from

3

to

201

,

53

is the

51^{\text{st}}

number counted. When counting backwards from

201

to

3

,

53

is the

n^{\text{th}}

number counted. What is

n

?

AMC 12·2013 · AMC 12B·Easy

Ray's car averages

40

miles per gallon of gasoline, and Tom's car averages

10

miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?

AMC 12·2013 · AMC 12B·Easy

The average age of

33

fifth-graders is

11

. The average age of

55

of their parents is

33

. What is the average age of all of these parents and fifth-graders?

AMC 12·2013 · AMC 12B·Easy

Real numbers

x

and

y

satisfy the equation

x^2 + y^2 = 10x - 6y - 34

. What is

x + y

?

AMC 12·2013 · AMC 12B·Easy

Jo and Blair take turns counting from

1

to one more than the last number said by the other person. Jo starts by saying

``1"

, so Blair follows by saying

``1, 2"

. Jo then says

``1, 2, 3"

, and so on. What is the

53^{\text{rd}}

number said?

AMC 12·2014 · AMC 12A·Easy

What is

10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?

AMC 12·2014 · AMC 12A·Medium

David drives from his home to the airport to catch a flight. He drives

35

miles in the first hour, but realizes that he will be

1

hour late if he continues at this speed. He increases his speed by

15

miles per hour for the rest of the way to the airport and arrives

30

minutes early. How many miles is the airport from his home?

AMC 12·2014 · AMC 12A·Medium

Let

a<b<c

be three integers such that

a,b,c

is an arithmetic progression and

a,c,b

is a geometric progression. What is the smallest possible value of

c

?

AMC 12·2014 · AMC 12A·Medium

The product

(8)(888\ldots 8)

, where the second factor has

k

digits, is an integer whose digits have a sum of

1000

. What is

k

?

AMC 12·2014 · AMC 12A·Medium

There are exactly

N

distinct rational numbers

k

such that

|k|<200

and

5x^2+kx+12=0

has at least one integer solution for

x

. What is

N

?

AMC 12·2014 · AMC 12A·Easy

At the theater children get in for half price. The price for

5

adult tickets and

4

child tickets is

\$24.50

. How much would

8

adult tickets and

6

child tickets cost?

AMC 12·2014 · AMC 12A·Hard

For every real number

x

, let

\lfloor x\rfloor

denote the greatest integer not exceeding

x

, and let

f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).

The set of all numbers

x

such that

1\leq x<2014

and

f(x)\leq 1

is a union of disjoint intervals. What is the sum of the lengths of those intervals?

AMC 12·2014 · AMC 12A·Hard

The fraction \dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0}, where

n

is the length of the period of the repeating decimal expansion. What is the sum

b_0+b_1+\cdots+b_{n-1}

?

AMC 12·2014 · AMC 12A·Hard

Let

f_0(x)=x+|x-100|-|x+100|

, and for

n\geq 1

, let

f_n(x)=|f_{n-1}(x)|-1

. For how many values of

x

is

f_{100}(x)=0

?

AMC 12·2014 · AMC 12A·Hard

The parabola

P

has focus

(0,0)

and goes through the points

(4,3)

and

(-4,-3)

. For how many points

(x,y)\in P

with integer coordinates is it true that

|4x+3y|\leq 1000

?

AMC 12·2014 · AMC 12A·Easy

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

AMC 12·2014 · AMC 12A·Easy

Suppose that

a

cows give

b

gallons of milk in

c

days. At this rate, how many gallons of milk will

d

cows give in

e

days?

AMC 12·2014 · AMC 12A·Easy

On an algebra quiz,

10\%

of the students scored

70

points,

35\%

scored

80

points,

30\%

scored

90

points, and the rest scored

100

points. What is the difference between the mean and median score of the students' scores on this quiz?

AMC 12·2014 · AMC 12A·Easy

The first three terms of a geometric progression are

\sqrt 3

,

\sqrt[3]3

, and

\sqrt[6]3

. What is the fourth term?

AMC 12·2014 · AMC 12A·Easy

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1:

10\%

off the listed price if the listed price is at least

\$50

Coupon 2:

\$20

off the listed price if the listed price is at least

\$100

Coupon 3:

18\%

off the amount by which the listed price exceeds

\$100

For which of the following listed prices will coupon

1

offer a greater price reduction than either coupon

2

or coupon

3

?

AMC 12·2014 · AMC 12A·Easy

Five positive consecutive integers starting with

a

have average

b

. What is the average of

5

consecutive integers that start with

b

?

AMC 12·2014 · AMC 12B·Easy

Leah has

13

coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?

AMC 12·2014 · AMC 12B·Medium

A list of

11

positive integers has a mean of

10

, a median of

9

, and a unique mode of

8

. What is the largest possible value of an integer in the list?

AMC 12·2014 · AMC 12B·Medium

A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?

AMC 12·2014 · AMC 12B·Medium

When

p = \sum\limits_{k=1}^{6} k \text{ ln }{k}

, the number

e^p

is an integer. What is the largest power of

2

that is a factor of

e^p

?

AMC 12·2014 · AMC 12B·Medium

Let

P

be a cubic polynomial with

P(0) = k

,

P(1) = 2k

, and

P(-1) = 3k

. What is

P(2) + P(-2)

?

AMC 12·2014 · AMC 12B·Medium

Let

P

be the parabola with equation

y=x^2

and let

Q = (20, 14)

. There are real numbers

r

and

s

such that the line through

Q

with slope

m

does not intersect

P

if and only if

r < m < s

. What is

r + s

?

AMC 12·2014 · AMC 12B·Easy

Orvin went to the store with just enough money to buy

30

balloons. When he arrived he discovered that the store had a special sale on balloons: buy

1

balloon at the regular price and get a second at

\frac{1}{3}

off the regular price. What is the greatest number of balloons Orvin could buy?

AMC 12·2014 · AMC 12B·Hard

Find the sum of all the positive solutions of

2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1

AMC 12·2014 · AMC 12B·Easy

Randy drove the first third of his trip on a gravel road, the next

20

miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?

AMC 12·2014 · AMC 12B·Easy

Susie pays for

4

muffins and

3

bananas. Calvin spends twice as much paying for

2

muffins and

16

bananas. A muffin is how many times as expensive as a banana?

AMC 12·2014 · AMC 12B·Easy

Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume

\frac{3}{4}

of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?

AMC 12·2015 · AMC 12A·Easy

What is the value of

(2^0-1+5^2-0)^{-1}\times5?

AMC 12·2015 · AMC 12A·Easy

Integers

x

and

y

with

x>y>0

satisfy

x+y+xy=80

. What is

x

?

AMC 12·2015 · AMC 12A·Medium

The parabolas

y=ax^2 - 2

and

y=4 - bx^2

intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area

12

. What is

a+b

?

AMC 12·2015 · AMC 12A·Medium

What is the value of

a

for which

\frac{1}{\log_2 a} + \frac{1}{\log_3 a} + \frac{1}{\log_4 a} = 1

?

AMC 12·2015 · AMC 12A·Medium

The zeros of the function

f(x) = x^2-ax+2a

are integers. What is the sum of the possible values of

a

?

AMC 12·2015 · AMC 12A·Easy

Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. After he graded Payton's test, the class average became 81. What was Payton's score on the test?

AMC 12·2015 · AMC 12A·Easy

The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?

AMC 12·2015 · AMC 12A·Easy

Sreshtha needs to estimate the quantity

\frac{a}{b} - c

, where

a, b,

and

c

are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of

\frac{a}{b} - c

?

AMC 12·2015 · AMC 12A·Easy

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be

2 : 1

?

AMC 12·2015 · AMC 12B·Easy

What is the value of

2-(-2)^{-2}

?

AMC 12·2015 · AMC 12B·Medium

Let

a

,

b

, and

c

be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation

(x-a)(x-b)+(x-b)(x-c)=0

?

AMC 12·2015 · AMC 12B·Easy

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

AMC 12·2015 · AMC 12B·Medium

For every positive integer

n

, let

\text{mod}_5 (n)

be the remainder obtained when

n

is divided by 5. Define a function

f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}

recursively as follows:

f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}

What is

f(2015,2)

?

AMC 12·2015 · AMC 12B·Easy

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?

AMC 12·2015 · AMC 12B·Easy

Lian, Marzuq, Rafsan, Arabi, Nabeel, and Rahul were in a 12-person race with 6 other people. Nabeel finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabeel. Arabi finished in 6th place. Who finished in 8th place?

AMC 12·2015 · AMC 12B·Easy

The Tigers beat the Sharks 2 out of the 3 times they played. They then played

N

more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for

N

?

AMC 12·2015 · AMC 12B·Easy

Back in 1930, Tillie had to memorize her multiplication facts from

0 \times 0

to

12 \times 12

. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?

AMC 12·2015 · AMC 12B·Easy

What is the value of

(625^{\log_5 2015})^{\frac{1}{4}}

?

AMC 12·2016 · AMC 12A·Easy

What is the value of

\frac{11!-10!}{9!}

?

AMC 12·2016 · AMC 12A·Easy

Five friends sat in a movie theater in a row containing

5

seats, numbered

1

to

5

from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

AMC 12·2016 · AMC 12A·Medium

Each of the

100

students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are

42

students who cannot sing,

65

students who cannot dance, and

29

students who cannot act. How many students have two of these talents?

AMC 12·2016 · AMC 12A·Medium

The graphs of

y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,

and

y=\log_x \dfrac{1}{3}

are plotted on the same set of axes. How many points in the plane with positive

x

-coordinates lie on two or more of the graphs?

AMC 12·2016 · AMC 12A·Medium

For some positive integer

n,

the number

110n^3

has

110

positive integer divisors, including

1

and the number

110n^3.

How many positive integer divisors does the number

81n^4

have?

AMC 12·2016 · AMC 12A·Easy

For what value of

x

does

10^x \cdot 100^{2x} = 1000^5

?

AMC 12·2016 · AMC 12A·Hard

There is a smallest positive real number

a

such that there exists a positive real number

b

such that all the roots of the polynomial

x^3-ax^2+bx-a

are real. In fact, for this value of

a

the value of

b

is unique. What is this value of

b

?

AMC 12·2016 · AMC 12A·Easy

The remainder can be defined for all real numbers

x

and

y

with

y \neq 0

by \text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor where

\left \lfloor \tfrac{x}{y} \right \rfloor

denotes the greatest integer less than or equal to

\tfrac{x}{y}

. What is the value of

\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )

?

AMC 12·2016 · AMC 12A·Easy

The mean, median, and mode of the

7

data values

60, 100, x, 40, 50, 200, 90

are all equal to

x

. What is the value of

x

?

AMC 12·2016 · AMC 12A·Easy

Which of these describes the graph of

x^2(x+y+1)=y^2(x+y+1)

?

AMC 12·2016 · AMC 12B·Easy

What is the value of

\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}

when

a= \frac{1}{2}

?

AMC 12·2016 · AMC 12B·Medium

All the numbers

1, 2, 3, 4, 5, 6, 7, 8, 9

are written in a

3\times3

array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to

18

. What is the number in the center?

AMC 12·2016 · AMC 12B·Medium

Alice and Bob live

10

miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is

30^\circ

from Alice's position and

60^\circ

from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

AMC 12·2016 · AMC 12B·Medium

The sum of an infinite geometric series is a positive number

S

, and the second term in the series is

1

. What is the smallest possible value of

S?

AMC 12·2016 · AMC 12B·Easy

The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of

1

and

2016

is closest to which integer?

AMC 12·2016 · AMC 12B·Hard

For a certain positive integer

n

less than

1000

, the decimal equivalent of

\frac{1}{n}

is

0.\overline{abcdef}

, a repeating decimal of period of

6

, and the decimal equivalent of

\frac{1}{n+6}

is

0.\overline{wxyz}

, a repeating decimal of period

4

. In which interval does

n

lie?

AMC 12·2016 · AMC 12B·Hard

The sequence

(a_n)

is defined recursively by

a_0=1

,

a_1=\sqrt[19]{2}

, and

a_n=a_{n-1}a_{n-2}^2

for

n\geq 2

. What is the smallest positive integer

k

such that the product

a_1a_2\cdots a_k

is an integer?

AMC 12·2016 · AMC 12B·Easy

Let

x=-2016

. What is the value of

\bigg| ||x|-x|-|x| \bigg| -x

?

AMC 12·2016 · AMC 12B·Easy

The ratio of the measures of two acute angles is

5:4

, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?

AMC 12·2016 · AMC 12B·Easy

The War of

1812

started with a declaration of war on Thursday, June

18

,

1812

. The peace treaty to end the war was signed

919

days later, on December

24

,

1814

. On what day of the week was the treaty signed?

AMC 12·2016 · AMC 12B·Easy

Josh writes the numbers

1,2,3,\dots,99,100

. He marks out

1

, skips the next number

(2)

, marks out

3

, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number

(2)

, skips the next number

(4)

, marks out

6

, skips

8

, marks out

10

, and so on to the end. Josh continues in this manner until only one number remains. What is that number?

AMC 12·2016 · AMC 12B·Easy

Carl decided to fence in his rectangular garden. He bought

20

fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly

4

yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl's garden?

AMC 12·2017 · AMC 12A·Easy

Pablo buys popsicles for his friends. The store sells single popsicles for

\$1

each, 3-popsicle boxes for

\$2

, and 5-popsicle boxes for

\$3

. What is the greatest number of popsicles that Pablo can buy with

\$8

?

AMC 12·2017 · AMC 12A·Medium

Driving at a constant speed, Sharon usually takes

180

minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving

\frac{1}{3}

of the way, she hits a bad snowstorm and reduces her speed by

20

miles per hour. This time the trip takes her a total of

276

minutes. How many miles is the drive from Sharon's house to her mother's house?

AMC 12·2017 · AMC 12A·Medium

Let

f(x) = \sin{x} + 2\cos{x} + 3\tan{x}

, using radian measure for the variable

x

. In what interval does the smallest positive value of

x

for which

f(x) = 0

lie?

AMC 12·2017 · AMC 12A·Medium

There are

24

different complex numbers

z

such that

z^{24}=1

. For how many of these is

z^6

a real number?

AMC 12·2017 · AMC 12A·Easy

The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?

AMC 12·2017 · AMC 12A·Medium

How many ordered pairs

(a,b)

such that

a

is a positive real number and

b

is an integer between

2

and

200

, inclusive, satisfy the equation

(\log_b a)^{2017}=\log_b(a^{2017})?

AMC 12·2017 · AMC 12A·Hard

A set

S

is constructed as follows. To begin,

S = \{0,10\}

. Repeatedly, as long as possible, if

x

is an integer root of some polynomial

a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0

for some

n\geq{1}

, all of whose coefficients

a_i

are elements of

S

, then

x

is put into

S

. When no more elements can be added to

S

, how many elements does

S

have?

AMC 12·2017 · AMC 12A·Hard

For certain real numbers

a

,

b

, and

c

, the polynomial

g(x) = x^3 + ax^2 + x + 10

has three distinct roots, and each root of

g(x)

is also a root of the polynomial

f(x) = x^4 + x^3 + bx^2 + 100x + c.

What is

f(1)

?

AMC 12·2017 · AMC 12A·Easy

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?

AMC 12·2017 · AMC 12A·Easy

At a gathering of

30

people, there are

20

people who all know each other and

10

people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?

AMC 12·2017 · AMC 12A·Easy

Define a function on the positive integers recursively by

f(1) = 2

,

f(n) = f(n-1) + 1

if

n

is even, and

f(n) = f(n-2) + 2

if

n

is odd and greater than

1

. What is

f(2017)

?

AMC 12·2017 · AMC 12A·Easy

Let

S

be the set of points

(x,y)

in the coordinate plane such that two of the three quantities

3

,

x+2

, and

y-4

are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of

S

?

AMC 12·2017 · AMC 12B·Easy

Kymbrea's comic book collection currently has

30

comic books in it, and she is adding to her collection at the rate of

2

comic books per month. LaShawn's collection currently has

10

comic books in it, and he is adding to his collection at the rate of

6

comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?

AMC 12·2017 · AMC 12B·Easy

At Typico High School,

60\%

of the students like dancing, and the rest dislike it. Of those who like dancing,

80\%

say that they like it, and the rest say that they dislike it. Of those who dislike dancing,

90\%

say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?

AMC 12·2017 · AMC 12B·Easy

Real numbers

x

,

y

, and

z

satify the inequalities

0<x<1

,

-1<y<0

, and

1<z<2

. Which of the following numbers is necessarily positive?

AMC 12·2017 · AMC 12B·Hard

Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?

AMC 12·2017 · AMC 12B·Hard

The graph of

y=f(x)

, where

f(x)

is a polynomial of degree

3

, contains points

A(2,4)

,

B(3,9)

, and

C(4,16)

. Lines

AB

,

AC

, and

BC

intersect the graph again at points

D

,

E

, and

F

, respectively, and the sum of the

x

-coordinates of

D

,

E

, and

F

is 24. What is

f(0)

?

AMC 12·2017 · AMC 12B·Easy

Supposed that

x

and

y

are nonzero real numbers such that

\frac{3x+y}{x-3y}=-2

. What is the value of

\frac{x+3y}{3x-y}

?

AMC 12·2017 · AMC 12B·Easy

Samia set off on her bicycle to visit her friend, traveling at an average speed of

17

kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at

5

kilometers per hour. In all it took her

44

minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?

AMC 12·2017 · AMC 12B·Easy

The data set

[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]

has median

Q_2 = 40

, first quartile

Q_1 = 33

, and third quartile

Q_3 = 43

. An outlier in a data set is a value that is more than

1.5

times the interquartile range below the first quartle (

Q_1

) or more than

1.5

times the interquartile range above the third quartile (

Q_3

), where the interquartile range is defined as

Q_3 - Q_1

. How many outliers does this data set have?

AMC 12·2017 · AMC 12B·Easy

The functions

\sin(x)

and

\cos(x)

are periodic with least period

2\pi

. What is the least period of the function

\cos(\sin(x))

?

AMC 12·2018 · AMC 12A·Easy

A large urn contains

100

balls, of which

36 \%

are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be

72 \%

? (No red balls are to be removed.)

AMC 12·2018 · AMC 12A·Easy

How many ordered pairs of real numbers

(x,y)

satisfy the following system of equations?

\begin{align*} x+3y&=3 \\ \big||x|-|y|\big|&=1 \end{align*}

AMC 12·2018 · AMC 12A·Medium

Let

S

be a set of

6

integers taken from

\{1,2,\dots,12\}

with the property that if

a

and

b

are elements of

S

with

a<b

, then

b

is not a multiple of

a

. What is the least possible value of an element in

S

?

AMC 12·2018 · AMC 12A·Medium

How many nonnegative integers can be written in the form a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0, where

a_i\in \{-1,0,1\}

for

0\le i \le 7

?

AMC 12·2018 · AMC 12A·Medium

Which of the following describes the set of values of

a

for which the curves

x^2+y^2=a^2

and

y=x^2-a

in the real

xy

-plane intersect at exactly

3

points?

AMC 12·2018 · AMC 12A·Easy

While exploring a cave, Carl comes across a collection of

5

-pound rocks worth

\$14

each,

4

-pound rocks worth

\$11

each, and

1

-pound rocks worth

\$2

each. There are at least

20

of each size. He can carry at most

18

pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

AMC 12·2018 · AMC 12A·Hard

Which of the following polynomials has the greatest real root?

AMC 12·2018 · AMC 12A·Easy

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least

6

miles away," Bob replied, "We are at most

5

miles away." Charlie then remarked, "Actually the nearest town is at most

4

miles away." It turned out that none of the three statements were true. Let

d

be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of

d

?

AMC 12·2018 · AMC 12A·Easy

What is the sum of all possible values of

k

for which the polynomials

x^2 - 3x + 2

and

x^2 - 5x + k

have a root in common?

AMC 12·2018 · AMC 12A·Easy

For positive integers

m

and

n

such that

m+10<n+1

, both the mean and the median of the set

\{m, m+4, m+10, n+1, n+2, 2n\}

are equal to

n

. What is

m+n

?

AMC 12·2018 · AMC 12A·Easy

For how many (not necessarily positive) integer values of

n

is the value of

4000\cdot \left(\tfrac{2}{5}\right)^n

an integer?

AMC 12·2018 · AMC 12A·Easy

Which of the following describes the largest subset of values of

y

within the closed interval

[0,\pi]

for which

\sin(x+y)\leq \sin(x)+\sin(y)

for every

x

between

0

and

\pi

, inclusive?

AMC 12·2018 · AMC 12B·Easy

Kate bakes a

20

-inch by

18

-inch pan of cornbread. The cornbread is cut into pieces that measure

2

inches by

2

inches. How many pieces of cornbread does the pan contain?

AMC 12·2018 · AMC 12B·Easy

A list of

2018

positive integers has a unique mode, which occurs exactly

10

times. What is the least number of distinct values that can occur in the list?

AMC 12·2018 · AMC 12B·Medium

Let

p

and

q

be positive integers such that \frac{5}{9} < \frac{p}{q} < \frac{4}{7} and

q

is as small as possible. What is

q-p

?

AMC 12·2018 · AMC 12B·Medium

A function

f

is defined recursively by

f(1)=f(2)=1

and f(n)=f(n-1)-f(n-2)+n for all integers

n \geq 3

. What is

f(2018)

?

AMC 12·2018 · AMC 12B·Easy

Sam drove

96

miles in

90

minutes. His average speed during the first

30

minutes was

60

mph (miles per hour), and his average speed during the second

30

minutes was

65

mph. What was his average speed, in mph, during the last

30

minutes?

AMC 12·2018 · AMC 12B·Hard

Consider polynomials

P(x)

of degree at most

3

, each of whose coefficients is an element of

\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}

. How many such polynomials satisfy

P(-1) = -9

?

AMC 12·2018 · AMC 12B·Hard

Let

\lfloor x \rfloor

denote the greatest integer less than or equal to

x

. How many real numbers

x

satisfy the equation

x^2 + 10,000\lfloor x \rfloor = 10,000x

?

AMC 12·2018 · AMC 12B·Easy

A line with slope

2

intersects a line with slope

6

at the point

(40,30)

. What is the distance between the

x

-intercepts of these two lines?

AMC 12·2018 · AMC 12B·Easy

Suppose

S

cans of soda can be purchased from a vending machine for

Q

quarters. Which of the following expressions describes the number of cans of soda that can be purchased for

D

dollars, where

1

dollar is worth

4

quarters?

AMC 12·2018 · AMC 12B·Easy

What is the value of

\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?

AMC 12·2018 · AMC 12B·Easy

What is

\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ?

AMC 12·2019 · AMC 12A·Medium

For some positive integer

k

, the repeating base-

k

representation of the (base-ten) fraction

\frac{7}{51}

is

0.\overline{23}_k = 0.232323..._k

. What is

k

?

AMC 12·2019 · AMC 12A·Medium

Positive real numbers

x \neq 1

and

y \neq 1

satisfy

\log_2{x} = \log_y{16}

and

xy = 64

. What is

(\log_2{\tfrac{x}{y}})^2

?

AMC 12·2019 · AMC 12A·Medium

For a certain complex number

c

, the polynomial

P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)

has exactly 4 distinct roots. What is

|c|

?

AMC 12·2019 · AMC 12A·Medium

Positive real numbers

a

and

b

have the property that \sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100 and all four terms on the left are positive integers, where

\log

denotes the base-

10

logarithm. What is

ab

?

AMC 12·2019 · AMC 12A·Medium

Let

s_k

denote the sum of the

\textit{k}

th powers of the roots of the polynomial

x^3-5x^2+8x-13

. In particular,

s_0=3

,

s_1=5

, and

s_2=9

. Let

a

,

b

, and

c

be real numbers such that

s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}

for

k = 2

,

3

,

....

What is

a+b+c

?

AMC 12·2019 · AMC 12A·Easy

Suppose

a

is

150\%

of

b

. What percent of

a

is

3b

?

AMC 12·2019 · AMC 12A·Hard

Define binary operations

\diamondsuit

and

\heartsuit

by

a \, \diamondsuit \, b = a^{\log_{7}(b)} \text{and} a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}

for all real numbers

a

and

b

for which these expressions are defined. The sequence

(a_n)

is defined recursively by

a_3 = 3\, \heartsuit\, 2

and

a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}

for all integers

n \geq 4

. To the nearest integer, what is

\log_{7}(a_{2019})

?

AMC 12·2019 · AMC 12A·Easy

What is the greatest number of consecutive integers whose sum is

45

?

AMC 12·2019 · AMC 12A·Easy

Melanie computes the mean

\mu

, the median

M

, and the modes of the

365

values that are the dates in the months of

2019

. Thus her data consist of

12 1\text{s}

,

12 2\text{s}

, . . . ,

12 28\text{s}

,

11 29\text{s}

,

11 30\text{s}

, and

7 31\text{s}

. Let

d

be the median of the modes. Which of the following statements is true?

AMC 12·2019 · AMC 12A·Easy

For a set of four distinct lines in a plane, there are exactly

N

distinct points that lie on two or more of the lines. What is the sum of all possible values of

N

?

AMC 12·2019 · AMC 12B·Hard

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is

ax^2+bx+c,a\neq 0,

and the roots are

r

and

s,

then the requirement is that

\{a,b,c\}=\{r,s\}

.)

AMC 12·2019 · AMC 12B·Hard

Define a sequence recursively by

x_0=5

and x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6} for all nonnegative integers

n.

Let

m

be the least positive integer such that x_m\leq 4+\frac{1}{2^{20}}. In which of the following intervals does

m

lie?

AMC 12·2019 · AMC 12B·Hard

How many sequences of

0

s and

1

s of length

19

are there that begin with a

0

, end with a

0

, contain no two consecutive

0

s, and contain no three consecutive

1

s?

AMC 12·2019 · AMC 12B·Easy

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either

12

pieces of red candy,

14

pieces of green candy,

15

pieces of blue candy, or

n

pieces of purple candy. A piece of purple candy costs

20

cents. What is the smallest possible value of

n

?

AMC 12·2019 · AMC 12B·Easy

What is the sum of all real numbers

x

for which the median of the numbers

4,6,8,17,

and

x

is equal to the mean of those five numbers?

AMC 12·2019 · AMC 12B·Easy

Let

f(x) = x^{2}(1-x)^{2}

. What is the value of the sum

f \left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots + f \left(\frac{2017}{2019} \right) - f \left(\frac{2018}{2019} \right)?

AMC 12·2020 · AMC 12A·Easy

Carlos took

70\%

of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?

AMC 12·2020 · AMC 12A·Medium

There are integers

a, b,

and

c,

each greater than

1,

such that

\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}

for all

N \neq 1

. What is

b

?

AMC 12·2020 · AMC 12A·Medium

In the complex plane, let

A

be the set of solutions to

z^3 - 8 = 0

and let

B

be the set of solutions to

z^3 - 8z^2 - 8z + 64 = 0

. What is the greatest distance between a point of

A

and a point of

B?

AMC 12·2020 · AMC 12A·Medium

There exists a unique strictly increasing sequence of nonnegative integers

a_1 < a_2 < … < a_k

such that \frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}. What is

k?

AMC 12·2020 · AMC 12A·Easy

The acronym AMC is shown in the rectangular grid below with grid lines spaced

1

unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC

?

AMC 12·2020 · AMC 12A·Hard

Let

(a_n)

and

(b_n)

be the sequences of real numbers such that

(2 + i)^n = a_n + b_ni

for all integers

n\geq 0

, where

i = \sqrt{-1}

. What is \sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?

AMC 12·2020 · AMC 12A·Easy

A driver travels for

2

hours at

60

miles per hour, during which her car gets

30

miles per gallon of gasoline. She is paid

\$0.50

per mile, and her only expense is gasoline at

\$2.00

per gallon. What is her net rate of pay, in dollars per hour, after this expense?

AMC 12·2020 · AMC 12A·Easy

The

25

integers from

-10

to

14,

inclusive, can be arranged to form a

5

-by-

5

square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

AMC 12·2020 · AMC 12A·Easy

In the plane figure shown below,

3

of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

AMC 12·2020 · AMC 12A·Easy

What is the median of the following list of

4040

numbers

? 1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2

AMC 12·2020 · AMC 12A·Easy

How many solutions does the equation

\tan{(2x)} = \cos{(\tfrac{x}{2})}

have on the interval

[0, 2\pi]?

AMC 12·2020 · AMC 12B·Easy

What is the value in simplest form of the following expression?

\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}

AMC 12·2020 · AMC 12B·Medium

Which of the following is the value of

\sqrt{\log_2{6}+\log_3{6}}?

AMC 12·2020 · AMC 12B·Medium

Bela and Jenn play the following game on the closed interval

[0, n]

of the real number line, where

n

is a fixed integer greater than

4

. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval

[0, n]

. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?

AMC 12·2020 · AMC 12B·Medium

How many polynomials of the form

x^5 + ax^4 + bx^3 + cx^2 + dx + 2020

, where

a

,

b

,

c

, and

d

are real numbers, have the property that whenever

r

is a root, so is

\frac{-1+i\sqrt{3}}{2} \cdot r

? (Note that

i=\sqrt{-1}

)

AMC 12·2020 · AMC 12B·Easy

What is the value of the following expression?

\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}

AMC 12·2020 · AMC 12B·Hard

How many positive integers

n

satisfy \frac{n+1000}{70} = \lfloor \sqrt{n} \rfloor? (Recall that

\lfloor x\rfloor

is the greatest integer not exceeding

x

.)

AMC 12·2020 · AMC 12B·Hard

What is the maximum value of

\frac{(2^t-3t)t}{4^t}

for real values of

t?

AMC 12·2020 · AMC 12B·Easy

The ratio of

w

to

x

is

4 : 3

, the ratio of

y

to

z

is

3 : 2

, and the ratio of

z

to

x

is

1 : 6

. What is the ratio of

w

to

y

?

AMC 12·2020 · AMC 12B·Easy

Teams

A

and

B

are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team

A

has won

\tfrac{2}{3}

of its games and team

B

has won

\tfrac{5}{8}

of its games. Also, team

B

has won

7

more games and lost

7

more games than team

A.

How many games has team

A

played?

AMC 12·2020 · AMC 12B·Easy

Two nonhorizontal, non vertical lines in the

xy

-coordinate plane intersect to form a

45^{\circ}

angle. One line has slope equal to

6

times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?

AMC 12·2020 · AMC 12B·Easy

How many ordered pairs of integers

(x, y)

satisfy the equation x^{2020}+y^2=2y?

AMC 12·2021 · AMC 12A·Easy

What is the value of

2^{1+2+3}-(2^1+2^2+2^3)?

AMC 12·2021 · AMC 12A·Medium

Of the following complex numbers

z

, which one has the property that

z^5

has the greatest real part?

AMC 12·2021 · AMC 12A·Medium

What is the value of

\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?

AMC 12·2021 · AMC 12A·Medium

How many solutions does the equation

\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)

have in the closed interval

[0,\pi]

?

AMC 12·2021 · AMC 12A·Easy

Under what conditions is

\sqrt{a^2+b^2}=a+b

true, where

a

and

b

are real numbers?

AMC 12·2021 · AMC 12A·Medium

Suppose that on a parabola with vertex

V

and a focus

F

there exists a point

A

such that

AF=20

and

AV=21

. What is the sum of all possible values of the length

FV?

AMC 12·2021 · AMC 12A·Hard

Suppose that the roots of the polynomial

P(x)=x^3+ax^2+bx+c

are

\cos \frac{2\pi}7,\cos \frac{4\pi}7,

and

\cos \frac{6\pi}7

, where angles are in radians. What is

abc

?

AMC 12·2021 · AMC 12A·Easy

A sequence of numbers is defined by

D_0=0,D_1=0,D_2=1

and

D_n=D_{n-1}+D_{n-3}

for

n\ge 3

. What are the parities (evenness or oddness) of the triple of numbers

(D_{2021},D_{2022},D_{2023})

, where

E

denotes even and

O

denotes odd?

AMC 12·2021 · AMC 12B·Medium

How many values of

\theta

in the interval

0<\theta\le 2\pi

satisfy

1-3\sin\theta+5\cos3\theta = 0?

AMC 12·2021 · AMC 12B·Medium

Let

g(x)

be a polynomial with leading coefficient

1,

whose three roots are the reciprocals of the three roots of

f(x)=x^3+ax^2+bx+c,

where

1<a<b<c.

What is

g(1)

in terms of

a,b,

and

c?

AMC 12·2021 · AMC 12B·Medium

Let

z

be a complex number satisfying

12|z|^2=2|z+2|^2+|z^2+1|^2+31.

What is the value of

z+\frac 6z?

AMC 12·2021 · AMC 12B·Hard

Let

S

be the sum of all positive real numbers

x

for which

x^{2^{\sqrt2}}=\sqrt2^{2^x}.

Which of the following statements is true?

AMC 12·2021 · AMC 12B·Easy

Suppose

2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.

What is the value of

x?

AMC 12·2021 · AMC 12B·Easy

What is the value of

\frac{\log_2 80}{\log_{40}2}-\frac{\log_2 160}{\log_{20}2}?

AMC 12·2021 · AMC 12A Fall·Medium

An organization has

30

employees,

20

of whom have a brand A computer while the other

10

have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?

AMC 12·2021 · AMC 12A Fall·Medium

Let

x

be the least real number greater than

1

such that

\sin(x)= \sin(x^2)

, where the arguments are in degrees. What is

x

rounded up to the closest integer?

AMC 12·2021 · AMC 12B Fall·Medium

Let

c = \frac{2\pi}{11}.

What is the value of

\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?

AMC 12·2021 · AMC 12B Fall·Medium

Suppose that

P(z), Q(z)

, and

R(z)

are polynomials with real coefficients, having degrees

2

,

3

, and

6

, respectively, and constant terms

1

,

2

, and

3

, respectively. Let

N

be the number of distinct complex numbers

z

that satisfy the equation

P(z) \cdot Q(z)=R(z)

. What is the minimum possible value of

N

?

AMC 12·2021 · AMC 12B Fall·Medium

Set

u_0 = \frac{1}{4}

, and for

k \ge 0

let

u_{k+1}

be determined by the recurrence

u_{k+1} = 2u_k - 2u_k^2.

This sequence tends to a limit; call it

L

. What is the least value of

k

such that

|u_k-L| \le \frac{1}{2^{1000}}?

AMC 12·2021 · AMC 12B Fall·Hard

For real numbers

x

, let

P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)

where

i = \sqrt{-1}

. For how many values of

x

with

0\leq x<2\pi

does

P(x)=0?

AMC 12·2022 · AMC 12A·Easy

What is the value of

3+\frac{1}{3+\frac{1}{3+\frac13}}?

AMC 12·2022 · AMC 12A·Medium

What is the product of all real numbers

x

such that the distance on the number line between

\log_6x

and

\log_69

is twice the distance on the number line between

\log_610

and

1

?

AMC 12·2022 · AMC 12A·Medium

What is the value of (\log 5)^3+(\log 20)^3+(\log 8)(\log 0.25) where

\log

denotes the base-ten logarithm?

AMC 12·2022 · AMC 12A·Medium

The roots of the polynomial

10x^3-39x^2+29x-6

are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by

2

units. What is the volume of the new box?

AMC 12·2022 · AMC 12A·Medium

Suppose

a

is a real number such that the equation

a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}

has more than one solution in the interval

(0, \pi)

. The set of all such

a

that can be written in the form

(p,q) \cup (q,r),

where

p, q,

and

r

are real numbers with

p < q< r

. What is

p+q+r

?

AMC 12·2022 · AMC 12A·Medium

Let

T_k

be the transformation of the coordinate plane that first rotates the plane

k

degrees counterclockwise around the origin and then reflects the plane across the

y

-axis. What is the least positive integer

n

such that performing the sequence of transformations

T_1, T_2, T_3, \dots, T_n

returns the point

(1,0)

back to itself?

AMC 12·2022 · AMC 12A·Medium

Suppose that

13

cards numbered

1, 2, 3, \ldots, 13

are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards

1, 2, 3

are picked up on the first pass,

4

and

5

on the second pass,

6

on the third pass,

7, 8, 9, 10

on the fourth pass, and

11, 12, 13

on the fifth pass. For how many of the

13!

possible orderings of the cards will the

13

cards be picked up in exactly two passes?

AMC 12·2022 · AMC 12A·Easy

The sum of three numbers is

96.

The first number is

6

times the third number, and the third number is

40

less than the second number. What is the absolute value of the difference between the first and second numbers?

AMC 12·2022 · AMC 12A·Hard

Let

P(x) = x^{2022} + x^{1011} + 1.

Which of the following polynomials is a factor of

P(x)

?

AMC 12·2022 · AMC 12A·Hard

How many strings of length

5

formed from the digits

0

,

1

,

2

,

3

,

4

are there such that for each

j \in \{1,2,3,4\}

, at least

j

of the digits are less than

j

? (For example,

02214

satisfies this condition because it contains at least

1

digit less than

1

, at least

2

digits less than

2

, at least

3

digits less than

3

, and at least

4

digits less than

4

. The string

23404

does not satisfy the condition because it does not contain at least

2

digits less than

2

.)

AMC 12·2022 · AMC 12A·Easy

The taxicab distance between points

(x_1, y_1)

and

(x_2, y_2)

in the coordinate plane is given by

|x_1 - x_2| + |y_1 - y_2|.

For how many points

P

with integer coordinates is the taxicab distance between

P

and the origin less than or equal to

20

?

AMC 12·2022 · AMC 12A·Easy

A data set consists of

6

(not distinct) positive integers:

1

,

7

,

5

,

2

,

5

, and

X

. The average (arithmetic mean) of the

6

numbers equals a value in the data set. What is the sum of all possible values of

X

?

AMC 12·2022 · AMC 12A·Easy

The infinite product

\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \cdots

evaluates to a real number. What is that number?

AMC 12·2022 · AMC 12A·Easy

On Halloween

31

children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the

22

children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the

15

children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the

9

children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth?

AMC 12·2022 · AMC 12B·Easy

Define

x\diamond y

to be

|x-y|

for all real numbers

x

and

y.

What is the value of

(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?

AMC 12·2022 · AMC 12B·Medium

Let

f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n

, where

i = \sqrt{-1}

. What is

f(2022)

?

AMC 12·2022 · AMC 12B·Medium

Suppose

x

and

y

are positive real numbers such that

x^y=2^{64}\text{ and }(\log_2{x})^{\log_2{y}}=2^{7}.

What is the greatest possible value of

\log_2{y}

?

AMC 12·2022 · AMC 12B·Medium

How many

4 \times 4

arrays whose entries are

0

s and

1

s are there such that the row sums (the sum of the entries in each row) are

1, 2, 3,

and

4,

in some order, and the column sums (the sum of the entries in each column) are also

1, 2, 3,

and

4,

in some order? For example, the array

\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} \right]

satisfies the condition.

AMC 12·2022 · AMC 12B·Medium

Each square in a

5 \times 5

grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules: * Any filled square with two or three filled neighbors remains filled. * Any empty square with exactly three filled neighbors becomes a filled square. * All other squares remain empty or become empty. A sample transformation is shown in the figure below. Suppose the

5 \times 5

grid has a border of empty squares surrounding a

3 \times 3

subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)

AMC 12·2022 · AMC 12B·Medium

Let

P(x)

be a polynomial with rational coefficients such that when

P(x)

is divided by the polynomial

x^2 + x + 1

, the remainder is

x + 2

, and when

P(x)

is divided by the polynomial

x^2 + 1

, the remainder is

2x + 1

. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

AMC 12·2022 · AMC 12B·Easy

For how many values of the constant

k

will the polynomial

x^{2}+kx+36

have two distinct integer roots?

AMC 12·2022 · AMC 12B·Easy

Consider the following

100

sets of

10

elements each:

\begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*}

How many of these sets contain exactly two multiples of

7

?

AMC 12·2022 · AMC 12B·Easy

Camila writes down five positive integers. The unique mode of these integers is

2

greater than their median, and the median is

2

greater than their arithmetic mean. What is the least possible value for the mode?

AMC 12·2022 · AMC 12B·Easy

What is the graph of

y^4+1=x^4+2y^2

in the coordinate plane?

AMC 12·2022 · AMC 12B·Easy

The sequence

a_0,a_1,a_2,\cdots

is a strictly increasing arithmetic sequence of positive integers such that

2^{a_7}=2^{27} \cdot a_7.

What is the minimum possible value of

a_2

?

AMC 12·2023 · AMC 12A·Easy

Cities

A

and

B

are

45

miles apart. Alicia lives in

A

and Beth lives in

B

. Alicia bikes towards

B

at 18 miles per hour. Leaving at the same time, Beth bikes toward

A

at 12 miles per hour. How many miles from City

A

will they be when they meet?

AMC 12·2023 · AMC 12A·Easy

Positive real numbers

x

and

y

satisfy

y^3=x^2

and

(y-x)^2=4y^2

. What is

x+y

?

AMC 12·2023 · AMC 12A·Medium

What is the degree measure of the acute angle formed by lines with slopes

2

and

\frac{1}{3}

?

AMC 12·2023 · AMC 12A·Medium

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was

40\%

more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

AMC 12·2023 · AMC 12A·Medium

How many complex numbers satisfy the equation

z^5=\overline{z}

, where

\overline{z}

is the conjugate of the complex number

z

?

AMC 12·2023 · AMC 12A·Medium

Consider the set of complex numbers

z

satisfying

|1+z+z^{2}|=4

. The maximum value of the imaginary part of

z

can be written in the form

\tfrac{\sqrt{m}}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

?

AMC 12·2023 · AMC 12A·Medium

What is the product of all solutions to the equation

\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023

AMC 12·2023 · AMC 12A·Easy

The weight of

\frac{1}{3}

of a large pizza together with

3 \frac{1}{2}

cups of orange slices is the same weight of

\frac{3}{4}

of a large pizza together with

\frac{1}{2}

cups of orange slices. A cup of orange slices weigh

\frac{1}{4}

of a pound. What is the weight, in pounds, of a large pizza?

AMC 12·2023 · AMC 12A·Hard

Let

f

be the unique function defined on the positive integers such that

\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1

for all positive integers

n

. What is

f(2023)

?

AMC 12·2023 · AMC 12A·Hard

How many ordered pairs of positive real numbers

(a,b)

satisfy the equation

(1+2a)(2+2b)(2a+b) = 32ab?

AMC 12·2023 · AMC 12A·Hard

There is a unique sequence of integers

a_1, a_2, \cdots a_{2023}

such that

\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}

whenever

\tan 2023x

is defined. What is

a_{2023}?

AMC 12·2023 · AMC 12A·Easy

How many digits are in the base-ten representation of

8^5 \cdot 5^{10} \cdot 15^5

?

AMC 12·2023 · AMC 12A·Easy

Points

A

and

B

lie on the graph of

y=\log_{2}x

. The midpoint of

\overline{AB}

is

(6, 2)

. What is the positive difference between the

x

-coordinates of

A

and

B

?

AMC 12·2023 · AMC 12A·Easy

A digital display shows the current date as an

8

-digit integer consisting of a

4

-digit year, followed by a

2

-digit month, followed by a

2

-digit date within the month. For example, Arbor Day this year is displayed as

20230428

. For how many dates in

2023

will each digit appear an even number of times in the 8-digital display for that date?

AMC 12·2023 · AMC 12A·Easy

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an

11

on the next quiz, her mean will increase by

1

. If she scores an

11

on each of the next three quizzes, her mean will increase by

2

. What is the mean of her quiz scores currently?

AMC 12·2023 · AMC 12B·Easy

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only

\frac{1}{3}

full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

AMC 12·2023 · AMC 12B·Medium

For complex number

u = a+bi

and

v = c+di

(where

i=\sqrt{-1}

), define the binary operation

u \otimes v = ac + bdi

Suppose

z

is a complex number such that

z\otimes z = z^{2}+40

. What is

|z|

?

AMC 12·2023 · AMC 12B·Medium

For how many ordered pairs

(a,b)

of integers does the polynomial

x^3+ax^2+bx+6

have

3

distinct integer roots?

AMC 12·2023 · AMC 12B·Medium

Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was

3

points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was

18

points higher than her average for the first semester and was again

3

points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?

AMC 12·2023 · AMC 12B·Easy

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by

20\%

on every pair of shoes. Carlos also knew that he had to pay a

7.5\%

sales tax on the discounted price. He had

43

dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

AMC 12·2023 · AMC 12B·Hard

A real-valued function

f

has the property that for all real numbers

a

and

b,

f(a + b) + f(a - b) = 2f(a) f(b).

Which one of the following cannot be the value of

f(1)?

AMC 12·2023 · AMC 12B·Easy

Jackson's paintbrush makes a narrow strip with a width of

6.5

millimeters. Jackson has enough paint to make a strip

25

meters long. How many square centimeters of paper could Jackson cover with paint?

AMC 12·2023 · AMC 12B·Easy

When the roots of the polynomial P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10} are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is

P(x)

positive?

AMC 12·2023 · AMC 12B·Easy

For how many integers

n

does the expression

\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}

represent a real number, where log denotes the base

10

logarithm?

AMC 12·2024 · AMC 12A·Easy

What is the value of

9901\cdot101-99\cdot10101?

AMC 12·2024 · AMC 12A·Medium

The first three terms of a geometric sequence are the integers

a,\,720,

and

b,

where

a<720<b.

What is the sum of the digits of the least possible value of

b?

AMC 12·2024 · AMC 12A·Medium

The numbers, in order, of each row and the numbers, in order, of each column of a

5 \times 5

array of integers form an arithmetic progression of length

5{.}

The numbers in positions

(5, 5), \,(2,4),\,(4,3),

and

(3, 1)

are

0, 48, 16,

and

12{,}

respectively. What number is in position

(1, 2)?

\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}

AMC 12·2024 · AMC 12A·Medium

The roots of

x^3 + 2x^2 - x + 3

are

p, q,

and

r.

What is the value of

(p^2 + 4)(q^2 + 4)(r^2 + 4)?

AMC 12·2024 · AMC 12A·Medium

Integers

a

,

b

, and

c

satisfy

ab + c = 100

,

bc + a = 87

, and

ca + b = 60

. What is

ab + bc + ca

?

AMC 12·2024 · AMC 12A·Medium

On top of a rectangular card with sides of length

1

and

2+\sqrt{3}

, an identical card is placed so that two of their diagonals line up, as shown (

\overline{AC}

, in this case). Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled

B

in the figure?

AMC 12·2024 · AMC 12A·Easy

Define

\blacktriangledown(a) = \sqrt{a - 1}

and

\blacktriangle(a) = \sqrt{a + 1}

for all real numbers

a

. What is the value of

\frac{\blacktriangledown(20 + \blacktriangle(2024))}{\blacktriangledown(\blacktriangle(24))} ?

AMC 12·2024 · AMC 12A·Hard

Suppose that

a_1 = 2

and the sequence

(a_n)

satisfies the recurrence relation

\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}

for all

n \ge 2.

What is the greatest integer less than or equal to

\sum^{100}_{n=1} a_n^2?

AMC 12·2024 · AMC 12A·Hard

What is the value of \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16} + \tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16} + \tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?

AMC 12·2024 · AMC 12A·Easy

What is the least value of

n

such that

n!

is a multiple of

2024

?

AMC 12·2024 · AMC 12A·Easy

A data set containing

20

numbers, some of which are

6

, has mean

45

. When all the 6s are removed, the data set has mean

66

. How many 6s were in the original data set?

AMC 12·2024 · AMC 12A·Easy

The product of three integers is

60

. What is the least possible positive sum of the three integers?

AMC 12·2024 · AMC 12A·Easy

How many angles

\theta

with

0\le\theta\le2\pi

satisfy

\log(\sin(3\theta))+\log(\cos(2\theta))=0

?

AMC 12·2024 · AMC 12B·Easy

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?

AMC 12·2024 · AMC 12B·Easy

A list of 9 real numbers consists of

1

,

2.2

,

3.2

,

5.2

,

6.2

, and

7

, as well as

x, y,z

with

x\leq y\leq z

. The range of the list is

7

, and the mean and median are both positive integers. How many ordered triples

(x,y,z)

are possible?

AMC 12·2024 · AMC 12B·Medium

Let

x_n = \sin^2(n^{\circ})

. What is the mean of

x_1,x_2,x_3,\dots,x_{90}

?

AMC 12·2024 · AMC 12B·Medium

There are real numbers

x,y,h

and

k

that satisfy the system of equations

x^2 + y^2 - 6x - 8y = h

x^2 + y^2 - 10x + 4y = k

What is the minimum possible value of

h+k

?

AMC 12·2024 · AMC 12B·Medium

Integers

a

and

b

are randomly chosen without replacement from the set of integers with absolute value not exceeding

10

. What is the probability that the polynomial

x^3 + ax^2 + bx + 6

has

3

distinct integer roots?

AMC 12·2024 · AMC 12B·Medium

The Fibonacci numbers are defined by

F_1=1, F_2=1,

and

F_n=F_{n-1}+F_{n-2}

for

n\geq 3.

What is \dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?

AMC 12·2024 · AMC 12B·Easy

What is

10! - 7! \cdot 6!

?

AMC 12·2024 · AMC 12B·Easy

For how many integer values of

x

is

|2x|\leq 7\pi?

AMC 12·2024 · AMC 12B·Easy

Balls numbered

1,2,3,\ldots

are deposited in

5

bins, labeled

A,B,C,D,

and

E

, using the following procedure. Ball

1

is deposited in bin

A

, and balls

2

and

3

are deposited in

B

. The next three balls are deposited in bin

C

, the next

4

in bin

D

, and so on, cycling back to bin

A

after balls are deposited in bin

E

. (For example,

22,23,\ldots,28

are deposited in bin

B

at step 7 of this process.) In which bin is ball

2024

deposited?

AMC 12·2024 · AMC 12B·Easy

In the following expression, Melanie changed some of the plus signs to minus signs: 1 + 3+5+7+\cdots+97+99 When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?

AMC 12·2024 · AMC 12B·Easy

The national debt of the United States is on track to reach

5 \cdot 10^{13}

dollars by

2033

. How many digits does this number of dollars have when written as a numeral in base

5

? (The approximation of

\log_{10} 5

as

0.7

is sufficient for this problem.)

AMC 12·2024 · AMC 12B·Easy

What value of

x

satisfies

\frac{\log_2x \cdot \log_3x}{\log_2x+\log_3x}=2?

AMC 12·2025 · AMC 12A·Easy

Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at

1{:}30

, traveling due north at a steady

8

miles per hour. Betsy leaves on her bicycle from the same point at

2{:}30

, traveling due east at a steady

12

miles per hour. At what time will they be exactly the same distance from their common starting point?

AMC 12·2025 · AMC 12A·Medium

The ''harmonic mean'' of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of

4,4,

and

5

is \frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}. What is the harmonic mean of all the real roots of the

4050

th degree polynomial

\prod_{k=1}^{2025} (kx^2-4x-3)=(x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)...(2025x^2-4x-3)?

AMC 12·2025 · AMC 12A·Medium

A set of numbers is called sum-free if whenever

x

and

y

are (not necessarily distinct) elements of the set,

x+y

is not an element of the set. For example,

\{1,4,6\}

and the empty set are sum-free, but

\{2,4,5\}

is not. What is the greatest possible number of elements in a sum-free subset of

\{1,2,3,...,20\}

?

AMC 12·2025 · AMC 12A·Medium

How many ordered triples

(x, y, z)

of different positive integers less than or equal to

8

satisfy

xy > z

,

xz > y

, and

yz > x

?

AMC 12·2025 · AMC 12A·Medium

Let

a

,

b

, and

c

be the roots of the polynomial

x^3 + kx + 1

. What is the sum

a^3b^2 + a^2b^3 + b^3c^2 + b^2c^3 + c^3a^2 + c^2a^3?

AMC 12·2025 · AMC 12A·Easy

A box contains

10

pounds of a nut mix that is

50

percent peanuts,

20

percent cashews, and

30

percent almonds. A second nut mix containing

20

percent peanuts,

40

percent cashews, and

40

percent almonds is added to the box resulting in a new nut mix that is

40

percent peanuts. How many pounds of cashews are now in the box?

AMC 12·2025 · AMC 12A·Hard

There is a unique ordered triple

(a,k,m)

of nonnegative integers such that \frac{4^a + 4^{a+k}+4^{a+2k}+\cdots + 4^{a+mk}}{2^a + 2^{a+k} + 2^{a+2k}+ \cdots + 2^{a+mk}} = 964. What is

a+k+m

?

AMC 12·2025 · AMC 12A·Easy

Agnes writes the following four statements on a blank piece of paper. *At least one of these statements is true. *At least two of these statements are true. *At least two of these statements are false. *At least one of these statements is false. Each statement is either true or false. How many false statements did Agnes write on the paper?

AMC 12·2025 · AMC 12A·Easy

Let

w

be the complex number

2+i

, where

i=\sqrt{-1}

. What real number

r

has the property that

r

,

w

, and

w^2

are three collinear points in the complex plane?

AMC 12·2025 · AMC 12B·Easy

The instructions on a

350

-gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires

20

grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?

AMC 12·2025 · AMC 12B·Medium

Consider a decreasing sequence of

n

positive integers

x_1>x_2>\cdots>x_n

that satisfies the following conditions: * The average of the first

3

terms in the sequence is

2025.

* For all

4 \leq k \leq n,

the average of the first

k

terms is

1

less than the average of the first

k - 1

terms. What is the greatest possible value of

{n?}

AMC 12·2025 · AMC 12B·Medium

An analog clock starts at midnight and runs for

2025

minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?

AMC 12·2025 · AMC 12B·Hard

How many real numbers satisfy the equation

\sin(20\pi x) = \log_{20}(x)

?

AMC 12·2025 · AMC 12B·Easy

What is the value of

i(i-1)(i-2)(i-3)

, where

i=\sqrt{-1}

?

AMC 12·2025 · AMC 12B·Easy

The value of the two-digit number

\underline{a}\;\underline{b}

in base seven equals the value of the two-digit number

\underline{b}\;\underline{a}

in base nine. What is

a+b

?

AMC 12·2025 · AMC 12B·Easy

Positive integers

x

and

y

satisfy the equation

57x+22y = 400

. What is the least possible value of

x+y

?

AMC 12·2025 · AMC 12B·Easy

Emmy says to Max, "I ordered 36 math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was

\$\underline{A}\;\underline{B}\;\underline{B}\;.\underline{B}\;\underline{A}

, where

A

and

B

are digits and

A \neq 0

." After a pause, Max says, "That was a good price." What is

A+B

?

AMC 12·2025 · AMC 12B·Easy

What is the value of \sum_{n=2}^{255} \frac{\log_2(1+\frac{1}{n})}{(\log_2 n)(\log_2(n+1))}

AMC 12·2025 · AMC 12B·Easy

There are integers

a

and

b

such that the polynomial

x^3 - 5x^2 + ax + b

has

4+\sqrt{5}

as a root. What is

a+b

?

Algebra