Mathletic | Practice Platform

AMC 12·2000 · AMC 12A·Easy

How many positive integers

b

have the property that

\log_{b} 729

is a positive integer?

AMC 12·2001 · AMC 12A·Medium

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?

AMC 12·2001 · AMC 12A·Medium

How many positive integers not exceeding

2001

are multiples of

3

or

4

but not

5

?

AMC 12·2001 · AMC 12A·Medium

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

AMC 12·2001 · AMC 12A·Medium

A point

P

is selected at random from the interior of the pentagon with vertices

A = (0,2)

,

B = (4,0)

,

C = (2 \pi + 1, 0)

,

D = (2 \pi + 1,4)

, and

E=(0,4)

. What is the probability that

\angle APB

is obtuse?

AMC 12·2001 · AMC 12A·Hard

Consider sequences of positive real numbers of the form

x, 2000, y, \dots

in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of

x

does the term

2001

appear somewhere in the sequence?

AMC 12·2002 · AMC 12A·Medium

Tina randomly selects two distinct numbers from the set

\{1, 2, 3, 4, 5\}

, and Sergio randomly selects a number from the set

\{1, 2, \ldots, 10\}

. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?

AMC 12·2002 · AMC 12A·Medium

Suppose that

a

and

b

are digits, not both nine and not both zero, and the repeating decimal

0.\overline{ab}

is expressed as a fraction in lowest terms. How many different denominators are possible?

AMC 12·2002 · AMC 12A·Hard

Triangle

ABC

is a right triangle with

\angle ACB

as its right angle,

m\angle ABC = 60^\circ

, and

AB = 10

. Let

P

be randomly chosen inside

ABC

, and extend

\overline{BP}

to meet

\overline{AC}

at

D

. What is the probability that

BD > 5\sqrt2

?

AMC 12·2002 · AMC 12A·Easy

For how many positive integers

m

does there exist at least one positive integer

n

such that

m \cdot n \le m + n

?

AMC 12·2002 · AMC 12B·Easy

How many different integers can be expressed as the sum of three distinct members of the set

\{1,4,7,10,13,16,19\}

?

AMC 12·2002 · AMC 12B·Medium

For how many integers

n

is

\dfrac n{20-n}

the square of an integer?

AMC 12·2002 · AMC 12B·Medium

Juan rolls a fair regular octahedral die marked with the numbers

1

through

8

. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?

AMC 12·2002 · AMC 12B·Medium

A point

P

is randomly selected from the rectangular region with vertices

(0,0),(2,0),(2,1),(0,1)

. What is the probability that

P

is closer to the origin than it is to the point

(3,1)

?

AMC 12·2002 · AMC 12 Practice·Easy

For how many positive integers

m

is

\frac{2002}{m^2 -2}

a positive integer?

AMC 12·2003 · AMC 12A·Medium

A point P is chosen at random in the interior of equilateral triangle

ABC

. What is the probability that

\triangle ABP

has a greater area than each of

\triangle ACP

and

\triangle BCP

?

AMC 12·2003 · AMC 12A·Hard

Objects

A

and

B

move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object

A

starts at

(0,0)

and each of its steps is either right or up, both equally likely. Object

B

starts at

(5,7)

and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

AMC 12·2003 · AMC 12A·Easy

What is the probability that a randomly drawn positive factor of

60

is less than

7

?

AMC 12·2003 · AMC 12B·Medium

Let

S

be the set of permutations of the sequence

1,2,3,4,5

for which the first term is not

1

. A permutation is chosen randomly from

S

. The probability that the second term is

2

, in lowest terms, is

a/b

. What is

a+b

?

AMC 12·2003 · AMC 12B·Hard

An object moves

8

cm in a straight line from

A

to

B

, turns at an angle

\alpha

, measured in radians and chosen at random from the interval

(0,\pi)

, and moves

5

cm in a straight line to

C

. What is the probability that

AC < 7

?

AMC 12·2003 · AMC 12B·Hard

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?

AMC 12·2004 · AMC 12A·Easy

For how many ordered pairs of positive integers

(x,y)

is

x+2y=100

?

AMC 12·2004 · AMC 12B·Medium

Each face of a cube is painted either red or blue, each with probability

1/2

. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

AMC 12·2005 · AMC 12A·Medium

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?

AMC 12·2005 · AMC 12A·Hard

Two distinct numbers a and b are chosen randomly from the set

\{2, 2^2, 2^3, ..., 2^{25}\}

. What is the probability that

\mathrm{\log}_a b

is an integer ?

AMC 12·2005 · AMC 12B·Medium

An envelope contains eight bills:

2

ones,

2

fives,

2

tens, and

2

twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $

20

or more?

AMC 12·2005 · AMC 12B·Hard

Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?

AMC 12·2006 · AMC 12A·Medium

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

AMC 12·2006 · AMC 12A·Hard

A circle of radius

r

is concentric with and outside a regular hexagon of side length

2

. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is

1/2

. What is

r

?

AMC 12·2006 · AMC 12A·Hard

How many non-empty subsets

S

of

\lbrace 1,2,3,\ldots ,15\rbrace

have the following two properties?

(1)

No two consecutive integers belong to

S

.

(2)

If

S

contains

k

elements, then

S

contains no number less than

k

.

AMC 12·2006 · AMC 12B·Medium

For a particular peculiar pair of dice, the probabilities of rolling

1

,

2

,

3

,

4

,

5

and

6

on each die are in the ratio

1:2:3:4:5:6

. What is the probability of rolling a total of

7

on the two dice?

AMC 12·2006 · AMC 12B·Medium

An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?

AMC 12·2006 · AMC 12B·Medium

Let

x

be chosen at random from the interval

(0,1)

. What is the probability that

\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0

? Here

\lfloor x\rfloor

denotes the greatest integer that is less than or equal to

x

.

AMC 12·2006 · AMC 12B·Hard

A sequence

a_1,a_2,\dots

of non-negative integers is defined by the rule

a_{n+2}=|a_{n+1}-a_n|

for

n\geq 1

. If

a_1=999

,

a_2<999

and

a_{2006}=1

, how many different values of

a_2

are possible?

AMC 12·2007 · AMC 12A·Medium

Integers

a, b, c,

and

d

, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that

ad-bc

is even?

AMC 12·2008 · AMC 12A·Medium

Let

a_1,a_2,\ldots

be a sequence determined by the rule

a_n=a_{n-1}/2

if

a_{n-1}

is even and

a_n=3a_{n-1}+1

if

a_{n-1}

is odd. For how many positive integers

a_1 \le 2008

is it true that

a_1

is less than each of

a_2

,

a_3

, and

a_4

?

AMC 12·2008 · AMC 12A·Hard

A permutation

(a_1,a_2,a_3,a_4,a_5)

of

(1,2,3,4,5)

is

\textit{heavy-tailed}

if

a_1 + a_2 < a_4 + a_5

. What is the number of heavy-tailed permutations?

AMC 12·2008 · AMC 12B·Easy

A basketball player made

5

baskets during a game. Each basket was worth either

2

or

3

points. How many different numbers could represent the total points scored by the player?

AMC 12·2008 · AMC 12B·Hard

Two circles of radius 1 are to be constructed as follows. The center of circle

A

is chosen uniformly and at random from the line segment joining

(0,0)

and

(2,0)

. The center of circle

B

is chosen uniformly and at random, and independently of the first choice, from the line segment joining

(0,1)

to

(2,1)

. What is the probability that circles

A

and

B

intersect?

AMC 12·2008 · AMC 12B·Hard

A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? (Solution)

AMC 12·2008 · AMC 12B·Easy

A class collects

50

dollars to buy flowers for a classmate who is in the hospital. Roses cost

3

dollars each, and carnations cost

2

dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly

50

dollars?

AMC 12·2009 · AMC 12A·Medium

How many positive integers less than

1000

are

6

times the sum of their digits?

AMC 12·2009 · AMC 12B·Medium

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?

AMC 12·2009 · AMC 12B·Medium

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every

90

seconds, and Robert runs clockwise and completes a lap every

80

seconds. Both start from the start line at the same time. At some random time between

10

minutes and

11

minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

AMC 12·2009 · AMC 12B·Hard

Ten women sit in

10

seats in a line. All of the

10

get up and then reseat themselves using all

10

seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?

AMC 12·2009 · AMC 12B·Hard

A region

S

in the complex plane is defined by S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}. A complex number

z = x + iy

is chosen uniformly at random from

S

. What is the probability that

\left(\frac34 + \frac34i\right)z

is also in

S

?

AMC 12·2009 · AMC 12B·Easy

By inserting parentheses, it is possible to give the expression 2\times3 + 4\times5 several values. How many different values can be obtained?

AMC 12·2010 · AMC 12A·Medium

A coin is altered so that the probability that it lands on heads is less than

\frac{1}{2}

and when the coin is flipped four times, the probability of an equal number of heads and tails is

\frac{1}{6}

. What is the probability that the coin lands on heads?

AMC 12·2010 · AMC 12A·Medium

Bernardo randomly picks 3 distinct numbers from the set

\{1,2,3,...,7,8,9\}

and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set

\{1,2,3,...,6,7,8\}

and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?

AMC 12·2010 · AMC 12A·Medium

A 16-step path is to go from

(-4,-4)

to

(4,4)

with each step increasing either the

x

-coordinate or the

y

-coordinate by 1. How many such paths stay outside or on the boundary of the square

-2 \le x \le 2

,

-2 \le y \le 2

at each step?

AMC 12·2010 · AMC 12A·Medium

Each of 2010 boxes in a line contains a single red marble, and for

1 \le k \le 2010

, the box in the

k\text{th}

position also contains

k

white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let

P(n)

be the probability that Isabella stops after drawing exactly

n

marbles. What is the smallest value of

n

for which

P(n) < \frac{1}{2010}

?

AMC 12·2010 · AMC 12B·Medium

For how many ordered triples

(x,y,z)

of nonnegative integers less than

20

are there exactly two distinct elements in the set

\{i^x, (1+i)^y, z\}

, where

i=\sqrt{-1}

?

AMC 12·2010 · AMC 12B·Medium

Positive integers

a

,

b

, and

c

are randomly and independently selected with replacement from the set

\{1, 2, 3,\dots, 2010\}

. What is the probability that

abc + ab + a

is divisible by

3

?

AMC 12·2010 · AMC 12B·Medium

A frog makes

3

jumps, each exactly

1

meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than

1

meter from its starting position?

AMC 12·2010 · AMC 12B·Easy

Every high school in the city of Euclid sent a team of

3

students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed

37

th and

64

th, respectively. How many schools are in the city?

AMC 12·2011 · AMC 12A·Easy

A pair of standard

6

-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

AMC 12·2011 · AMC 12A·Medium

Suppose

a

and

b

are single-digit positive integers chosen independently and at random. What is the probability that the point

(a,b)

lies above the parabola

y=ax^2-bx

?

AMC 12·2011 · AMC 12A·Medium

Each vertex of convex polygon

ABCDE

is to be assigned a color. There are

6

colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?

AMC 12·2011 · AMC 12B·Medium

A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

AMC 12·2011 · AMC 12B·Hard

For every

m

and

k

integers with

k

odd, denote by

\left[\frac{m}{k}\right]

the integer closest to

\frac{m}{k}

. For every odd integer

k

, let

P(k)

be the probability that

\left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right]

for an integer

n

randomly chosen from the interval

1 \leq n \leq 99!

. What is the minimum possible value of

P(k)

over the odd integers

k

in the interval

1 \leq k \leq 99

?

AMC 12·2011 · AMC 12B·Easy

Two real numbers are selected independently and at random from the interval

[-20,10]

. What is the probability that the product of those numbers is greater than zero?

AMC 12·2012 · AMC 12A·Medium

Alex, Mel, and Chelsea play a game that has

6

rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is

\frac{1}{2}

, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?

AMC 12·2012 · AMC 12A·Medium

A

3\times3

square is partitioned into

9

unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated

90^\circ

clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?

AMC 12·2012 · AMC 12A·Medium

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?

AMC 12·2012 · AMC 12A·Hard

Let

S

be the square one of whose diagonals has endpoints

(1/10,7/10)

and

(-1/10,-7/10)

. A point

v=(x,y)

is chosen uniformly at random over all pairs of real numbers

x

and

y

such that

0 \le x \le 2012

and

0\le y\le 2012

. Let

T(v)

be a translated copy of

S

centered at

v

. What is the probability that the square region determined by

T(v)

contains exactly two points with integer coefficients in its interior?

AMC 12·2012 · AMC 12B·Medium

How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?

AMC 12·2012 · AMC 12B·Medium

Two parabolas have equations

y= x^2 + ax +b

and

y= x^2 + cx +d

, where

a, b, c,

and

d

are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?

AMC 12·2012 · AMC 12B·Medium

Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?

AMC 12·2012 · AMC 12B·Hard

A bug travels from

A

to

B

along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

AMC 12·2012 · AMC 12B·Easy

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?

AMC 12·2013 · AMC 12A·Medium

Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?

AMC 12·2013 · AMC 12A·Medium

A group of

12

pirates agree to divide a treasure chest of gold coins among themselves as follows. The

k^\text{th}

pirate to take a share takes

\frac{k}{12}

of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the

12^{\text{th}}

pirate receive?

AMC 12·2013 · AMC 12A·Medium

Let

S

be the set

\{1,2,3,...,19\}

. For

a,b \in S

, define

a \succ b

to mean that either

0 < a - b \le 9

or

b - a > 9

. How many ordered triples

(x,y,z)

of elements of

S

have the property that

x \succ y

,

y \succ z

, and

z \succ x

?

AMC 12·2013 · AMC 12A·Hard

A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome

n

is chosen uniformly at random. What is the probability that

\frac{n}{11}

is also a palindrome?

AMC 12·2013 · AMC 12A·Hard

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular

12

-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?

AMC 12·2013 · AMC 12B·Medium

Cities

A

,

B

,

C

,

D

, and

E

are connected by roads

\widetilde{AB}

,

\widetilde{AD}

,

\widetilde{AE}

,

\widetilde{BC}

,

\widetilde{BD}

,

\widetilde{CD}

, and

\widetilde{DE}

. How many different routes are there from

A

to

B

that use each road exactly once? (Such a route will necessarily visit some cities more than once.)

AMC 12·2013 · AMC 12B·Medium

Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove

2

or

4

coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove

1

or

3

coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with

2013

coins and when the game starts with

2014

coins?

AMC 12·2014 · AMC 12A·Medium

A fancy bed and breakfast inn has

5

rooms, each with a distinctive color-coded decor. One day

5

friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than

2

friends per room. In how many ways can the innkeeper assign the guests to the rooms?

AMC 12·2014 · AMC 12A·Hard

The number

5^{867}

is between

2^{2013}

and

2^{2014}

. How many pairs of integers

(m,n)

are there such that

1\leq m\leq 2012

and 5^n<2^m<2^{m+2}<5^{n+1}?

AMC 12·2014 · AMC 12B·Medium

For how many positive integers

x

is

\log_{10}(x-40) + \log_{10}(60-x) < 2

?

AMC 12·2014 · AMC 12B·Hard

In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad

N

,

0<N<10

, it will jump to pad

N-1

with probability

\frac{N}{10}

and to pad

N+1

with probability

1-\frac{N}{10}

. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?

AMC 12·2014 · AMC 12B·Easy

For how many positive integers

n

is

\frac{n}{30-n}

also a positive integer?

AMC 12·2014 · AMC 12B·Easy

In the addition shown below

A

,

B

,

C

, and

D

are distinct digits. How many different values are possible for

D

?

\begin{array}{c}&A&B&B&C&B\\ +&B&C&A&D&A\\ \hline &D&B&D&D&D\end{array}

AMC 12·2015 · AMC 12A·Medium

On a sheet of paper, Isabella draws a circle of radius

2

, a circle of radius

3

, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly

k \ge 0

lines. How many different values of

k

are possible?

AMC 12·2015 · AMC 12A·Medium

A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?

AMC 12·2015 · AMC 12A·Medium

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

AMC 12·2015 · AMC 12A·Hard

Rational numbers

a

and

b

are chosen at random among all rational numbers in the interval

[0,2)

that can be written as fractions

\frac{n}{d}

where

n

and

d

are integers with

1 \le d \le 5

. What is the probability that

(\text{\cos}(a\pi)+i\text{\sin}(b\pi))^4

is a real number?

AMC 12·2015 · AMC 12A·Easy

A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?

AMC 12·2015 · AMC 12B·Medium

At Rachelle's school, an A counts

4

points, a B

3

points, a C

2

points, and a D

1

point. Her GPA in the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get A's in both Mathematics and Science and at least a C in each of English and History. She thinks she has a

\frac{1}{6}

chance of getting an A in English, and a

\tfrac{1}{4}

chance of getting a B. In History, she has a

\frac{1}{4}

chance of getting an A, and a

\frac{1}{3}

chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least

3.5

?

AMC 12·2015 · AMC 12B·Medium

An unfair coin lands on heads with a probability of

\tfrac{1}{4}

. When tossed

n

times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of

n

?

AMC 12·2015 · AMC 12B·Hard

Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?

AMC 12·2015 · AMC 12B·Easy

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is

\tfrac{1}{2}

, independently of what has happened before. What is the probability that Larry wins the game?

AMC 12·2016 · AMC 12A·Medium

Let

N

be a positive multiple of

5

. One red ball and

N

green balls are arranged in a line in random order. Let

P(N)

be the probability that at least

\tfrac{3}{5}

of the green balls are on the same side of the red ball. Observe that

P(5)=1

and that

P(N)

approaches

\tfrac{4}{5}

as

N

grows large. What is the sum of the digits of the least value of

N

such that

P(N) < \tfrac{321}{400}

?

AMC 12·2016 · AMC 12A·Medium

Each vertex of a cube is to be labeled with an integer from

1

through

8

, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?

AMC 12·2016 · AMC 12A·Medium

Jerry starts at

0

on the real number line. He tosses a fair coin

8

times. When he gets heads, he moves

1

unit in the positive direction; when he gets tails, he moves

1

unit in the negative direction. The probability that he reaches

4

at some time during this process

\frac{a}{b},

where

a

and

b

are relatively prime positive integers. What is

a + b?

(For example, he succeeds if his sequence of tosses is

HTHHHHHH.

)

AMC 12·2016 · AMC 12A·Hard

Three numbers in the interval

\left[0,1\right]

are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?

AMC 12·2016 · AMC 12B·Medium

In how many ways can

345

be written as the sum of an increasing sequence of two or more consecutive positive integers?

AMC 12·2016 · AMC 12B·Medium

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?

AMC 12·2016 · AMC 12B·Medium

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won

10

games and lost

10

games; there were no ties. How many sets of three teams

\{A, B, C\}

were there in which

A

beat

B

,

B

beat

C

, and

C

beat

A?

AMC 12·2017 · AMC 12A·Easy

Chloe chooses a real number uniformly at random from the interval

[ 0,2017 ]

. Independently, Laurent chooses a real number uniformly at random from the interval

[ 0 , 4034 ]

. What is the probability that Laurent's number is greater than Chloe's number?

AMC 12·2017 · AMC 12A·Medium

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of

5

chairs under these conditions?

AMC 12·2017 · AMC 12A·Hard

A square is drawn in the Cartesian coordinate plane with vertices at

(2, 2)

,

(-2, 2)

,

(-2, -2)

,

(2, -2)

. A particle starts at

(0,0)

. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is

1/8

that the particle will move from

(x, y)

to each of

(x, y + 1)

,

(x + 1, y + 1)

,

(x + 1, y)

,

(x + 1, y - 1)

,

(x, y - 1)

,

(x - 1, y - 1)

,

(x - 1, y)

, or

(x - 1, y + 1)

. The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is

m/n

, where

m

and

n

are relatively prime positive integers. What is

m + n

?

AMC 12·2017 · AMC 12A·Hard

The vertices

V

of a centrally symmetric hexagon in the complex plane are given by

V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.

For each

j

,

1\leq j\leq 12

, an element

z_j

is chosen from

V

at random, independently of the other choices. Let

P={\prod}_{j=1}^{12}z_j

be the product of the

12

numbers selected. What is the probability that

P=-1

?

AMC 12·2017 · AMC 12B·Medium

In the figure below,

3

of the

6

disks are to be painted blue,

2

are to be painted red, and

1

is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

AMC 12·2017 · AMC 12B·Medium

The number

21!=51,090,942,171,709,440,000

has over

60,000

positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

AMC 12·2017 · AMC 12B·Medium

A coin is biased in such a way that on each toss the probability of heads is

\frac{2}{3}

and the probability of tails is

\frac{1}{3}

. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?

AMC 12·2017 · AMC 12B·Medium

Real numbers

x

and

y

are chosen independently and uniformly at random from the interval

(0,1)

. What is the probability that

\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor

?

AMC 12·2017 · AMC 12B·Hard

Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?

AMC 12·2018 · AMC 12A·Hard

Alice, Bob, and Carol play a game in which each of them chooses a real number between

0

and

1.

The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between

0

and

1,

and Bob announces that he will choose his number uniformly at random from all the numbers between

\tfrac{1}{2}

and

\tfrac{2}{3}.

Armed with this information, what number should Carol choose to maximize her chance of winning?

AMC 12·2018 · AMC 12A·Easy

How many ways can a student schedule

3

mathematics courses -- algebra, geometry, and number theory -- in a

6

-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other

3

periods is of no concern here.)

AMC 12·2019 · AMC 12A·Medium

How many ways are there to paint each of the integers

2, 3, \cdots , 9

either red, green, or blue so that each number has a different color from each of its proper divisors?

AMC 12·2019 · AMC 12A·Medium

The numbers

1,2,\dots,9

are randomly placed into the

9

squares of a

3 \times 3

grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

AMC 12·2019 · AMC 12A·Medium

Real numbers between

0

and

1

, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is

0

if the second flip is heads and

1

if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval

[0,1]

. Two random numbers

x

and

y

are chosen independently in this manner. What is the probability that

|x-y| > \tfrac{1}{2}

?

AMC 12·2019 · AMC 12A·Hard

For how many integers

n

between

1

and

50

, inclusive, is \frac{(n^2-1)!}{(n!)^n} an integer? (Recall that

0! = 1

.)

AMC 12·2019 · AMC 12A·Easy

A box contains

28

red balls,

20

green balls,

19

yellow balls,

13

blue balls,

11

white balls, and

9

black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least

15

balls of a single color will be drawn?

AMC 12·2019 · AMC 12B·Easy

The figure below is a map showing

12

cities and

17

roads connecting certain pairs of cities. Paula wishes to travel along exactly

13

of those roads, starting at city

A

and ending at city

L

, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.) How many different routes can Paula take?

AMC 12·2019 · AMC 12B·Medium

A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin

k

is

2^{-k}

for

k = 1,2,3....

What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

AMC 12·2019 · AMC 12B·Medium

There are lily pads in a row numbered

0

to

11

, in that order. There are predators on lily pads

3

and

6

, and a morsel of food on lily pad

10

. Fiona the frog starts on pad

0

, and from any given lily pad, has a

\frac{1}{2}

chance to hop to the next pad, and an equal chance to jump

2

pads. What is the probability that Fiona reaches pad

10

without landing on either pad

3

or pad

6

?

AMC 12·2019 · AMC 12B·Medium

Raashan, Sylvia, and Ted play the following game. Each starts with

\$1

. A bell rings every

15

seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives

\$1

to that player. What is the probability that after the bell has rung

2019

times, each player will have

\$1

? (For example, Raashan and Ted may each decide \to give

\$1

to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have

\$0

, Sylvia will have

\$2

, and Ted will have

\$1

, and that is the \end of the first round of play. In the second round Rashaan has no money \to give, but Sylvia and Ted might choose each other \to give their

\$1

to, and the holdings will be the same at the end of the second round.)

AMC 12·2020 · AMC 12A·Medium

A frog sitting at the point

(1, 2)

begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length

1

, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices

(0,0), (0,4), (4,4),

and

(4,0)

. What is the probability that the sequence of jumps ends on a vertical side of the square

?

AMC 12·2020 · AMC 12A·Medium

A point is chosen at random within the square in the coordinate plane whose vertices are

(0, 0), (2020, 0), (2020, 2020),

and

(0, 2020)

. The probability that the point is within

d

units of a lattice point is

\tfrac{1}{2}

. (A point

(x, y)

is a lattice point if

x

and

y

are both integers.) What is

d

to the nearest tenth

?

AMC 12·2020 · AMC 12A·Hard

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly

7

. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

AMC 12·2020 · AMC 12B·Medium

There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?

AMC 12·2020 · AMC 12B·Medium

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?

AMC 12·2020 · AMC 12B·Medium

Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?

AMC 12·2020 · AMC 12B·Hard

Let

D(n)

denote the number of ways of writing the positive integer

n

as a product n = f_1\cdot f_2\cdots f_k, where

k\ge1

, the

f_i

are integers strictly greater than

1

, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number

6

can be written as

6

,

2\cdot 3

, and

3\cdot2

, so

D(6) = 3

. What is

D(96)

?

AMC 12·2020 · AMC 12B·Hard

For each real number

a

with

0 \leq a \leq 1

, let numbers

x

and

y

be chosen independently at random from the intervals

[0, a]

and

[0, 1]

, respectively, and let

P(a)

be the probability that

\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1

What is the maximum value of

P(a)?

AMC 12·2021 · AMC 12A·Easy

A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is

\frac13

. When

4

black cards are added to the deck, the probability of choosing red becomes

\frac14

. How many cards were in the deck originally?

AMC 12·2021 · AMC 12B·Medium

Two fair dice, each with at least

6

faces are rolled. On each face of each die is printed a distinct integer from

1

to the number of faces on that die, inclusive. The probability of rolling a sum of

7

is

\frac34

of the probability of rolling a sum of

10,

and the probability of rolling a sum of

12

is

\frac{1}{12}

. What is the least possible number of faces on the two dice combined?

AMC 12·2021 · AMC 12B·Hard

Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin

i

is

2^{-i}

for

i=1,2,3,....

More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is

\frac pq,

where

p

and

q

are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins

3,17,

and

10.

) What is

p+q?

AMC 12·2021 · AMC 12A Fall·Medium

What is the number of terms with rational coefficients among the

1001

terms in the expansion of

\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?

AMC 12·2021 · AMC 12A Fall·Hard

Azar and Carl play a game of tic-tac-toe. Azar places an

X

one of the boxes in a

3

-by-

3

array of boxes, then Carl places an

O

in one of the remaining boxes. After that, Azar places an

X

in one of the remaining boxes, and so on until all 9 boxes are filled or one of the players has 3 of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third

O

. How many ways can the board look after the game is over?

AMC 12·2021 · AMC 12B Fall·Medium

A bug starts at a vertex of a grid made of equilateral triangles of side length

1

. At each step the bug moves in one of the

6

possible directions along the grid lines randomly and independently with equal probability. What is the probability that after

5

moves the bug never will have been more than

1

unit away from the starting position?

AMC 12·2021 · AMC 12B Fall·Hard

What is the average number of pairs of consecutive integers in a randomly selected subset of

5

distinct integers chosen from the set

\{ 1, 2, 3, …, 30\}

? (For example the set

\{1, 17, 18, 19, 30\}

has

2

pairs of consecutive integers.)

AMC 12·2022 · AMC 12A·Easy

How many ways are there to split the integers

1

through

14

into

7

pairs such that in each pair, the greater number is at least

2

times the lesser number?

AMC 12·2022 · AMC 12A·Easy

A rectangle is partitioned into

5

regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?

AMC 12·2022 · AMC 12B·Medium

Kayla rolls four fair

6

-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than

4

and at least two of the numbers she rolls are greater than

2

?

AMC 12·2022 · AMC 12B·Hard

Ant Amelia starts on the number line at

0

and crawls in the following manner. For

n=1,2,3,

Amelia chooses a time duration

t_n

and an increment

x_n

independently and uniformly at random from the interval

(0,1).

During the

n

th step of the process, Amelia moves

x_n

units in the positive direction, using up

t_n

minutes. If the total elapsed time has exceeded

1

minute during the

n

th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most

3

steps in all. What is the probability that Amelia’s position when she stops will be greater than

1

?

AMC 12·2023 · AMC 12A·Medium

Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance

m

with probability

\frac{1}{2^m}

. What is the probability that Flora will eventually land at 10?

AMC 12·2023 · AMC 12A·Hard

If

A

and

B

are vertices of a polyhedron, define the distance

d(A, B)

to be the minimum number of edges of the polyhedron one must traverse in order to connect

A

and

B

. For example, if

\overline{AB}

is an edge of the polyhedron, then

d(A, B) = 1

, but if

\overline{AC}

and

\overline{CB}

are edges and

\overline{AB}

is not an edge, then

d(A, B) = 2

. Let

Q

,

R

, and

S

be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that

d(Q, R) > d(R, S)

?

AMC 12·2023 · AMC 12A·Easy

Janet rolls a standard

6

-sided die

4

times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal

3?

AMC 12·2023 · AMC 12B·Medium

Each of

2023

balls is placed in one of

3

bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

AMC 12·2023 · AMC 12B·Medium

Cyrus the frog jumps

2

units in a direction, then

2

more in another direction. What is the probability that he lands less than

1

unit away from his starting position?

AMC 12·2023 · AMC 12B·Hard

When

n

standard six-sided dice are rolled, the product of the numbers rolled can be any of

936

possible values. What is

n

?

AMC 12·2023 · AMC 12B·Easy

How many nonempty subsets

B

of

\{0, 1, 2, 3, \cdots, 12\}

have the property that the number of elements in

B

is equal to the least element of

B

? For example,

B = \{4, 6, 8, 11\}

satisfies the condition.

AMC 12·2024 · AMC 12A·Medium

A set of

12

tokens —

3

red,

2

white,

1

blue, and

6

black — is to be distributed at random to

3

game players,

4

tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

?

AMC 12·2024 · AMC 12A·Medium

Points

P

and

Q

are chosen uniformly and independently at random on sides

\overline {AB}

and

\overline{AC},

respectively, of equilateral triangle

\triangle ABC.

Which of the following intervals contains the probability that the area of

\triangle APQ

is less than half the area of

\triangle ABC?

AMC 12·2024 · AMC 12A·Hard

The figure below shows a dotted grid

8

cells wide and

3

cells tall consisting of

1''\times1''

squares. Carl places

1

-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?

AMC 12·2024 · AMC 12B·Medium

How many different remainders can result when the

100

th power of an integer is divided by

125

?

AMC 12·2024 · AMC 12B·Medium

A group of

16

people will be partitioned into

4

indistinguishable

4

-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as

3^{r}M

, where

r

and

M

are positive integers and

M

is not divisible by

3

. What is

r

?

AMC 12·2024 · AMC 12B·Hard

Pablo will decorate each of

6

identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the

12

decisions he must make. After the paint dries, he will place the

6

balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as

\frac mn,

where

m

and

n

are relatively prime positive integers. What is

m?

(Recall that two events

A

and

B

are independent if

P(A \text{ and }B) = P(A) \cdot P(B).

)

AMC 12·2025 · AMC 12A·Medium

Let

C = \{1, 2, 3, \dots, 13\}

. Let

N

be the greatest integer such that there exists a subset of

C

with

N

elements that does not contain five consecutive integers. Suppose

N

integers are chosen at random from

C

without replacement. What is the probability that the chosen elements do not include five consecutive integers?

AMC 12·2025 · AMC 12A·Hard

Three real numbers are chosen independently and uniformly at random between

0

and

1

. What is the probability that the greatest of these three numbers is greater than

2

times each of the other two numbers? (In other words, if the chosen numbers are

a \geq b \geq c

, then

a > 2b

.)

AMC 12·2025 · AMC 12A·Easy

A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is

15

. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from

12

to

14

. If Ash plays with the teachers, the average age on that team will decrease from

55

to

52

. How old is Ash?

AMC 12·2025 · AMC 12A·Easy

Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?

AMC 12·2025 · AMC 12B·Medium

Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing 3 blue bands, 3 red bands, and 3 green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?

AMC 12·2025 · AMC 12B·Medium

A circle has been divided into 6 sectors of different sizes. Then 2 of the sectors are painted red, 2 painted green, and 2 painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below. How many different colorings are possible?

AMC 12·2025 · AMC 12B·Medium

Each of the

9

squares in a

{3 \times 3}

grid is to be colored red, blue, or yellow in such a way that each red square shares an edge with at least one blue square, each blue square shares an edge with at least one yellow square, and each yellow square shares an edge with at least one red square. Colorings that can be obtained from one another by rotations and/or reflections are considered the same. How many different colorings are possible?

AMC 12·2025 · AMC 12B·Medium

Awnik repeatedly plays a game that has a probability of winning of

\frac{1}{3}

. The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?

AMC 12·2025 · AMC 12B·Medium

A frog hops along the number line according to the following rules: * It starts at

0

. * If it is at

0

, then it moves to

1

with probability

\frac 12

and disappears with probability

\frac 12

. * For

n=1,2,3

, if it is at

n

, then it moves to

n+1

with probability

\frac 14

, to

n-1

with probability

\frac 14

, and disappears with probability

\frac 12.{}

What is the probability that the frog reaches

4

?

Counting & Probability