Mathletic | Practice Platform

AMC 12·2001 · AMC 12A·Medium

Given the nine-sided regular polygon

A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9

, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set

\{A_1,A_2,\dots,A_9\}

?

AMC 12·2001 · AMC 12A·Medium

An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)

AMC 12·2001 · AMC 12A·Medium

A circle centered at

A

with a radius of 1 and a circle centered at

B

with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is

AMC 12·2001 · AMC 12A·Medium

Points

A = (3,9)

,

B = (1,1)

,

C = (5,3)

, and

D=(a,b)

lie in the first quadrant and are the vertices of quadrilateral

ABCD

. The quadrilateral formed by joining the midpoints of

\overline{AB}

,

\overline{BC}

,

\overline{CD}

, and

\overline{DA}

is a square. What is the sum of the coordinates of point

D

?

AMC 12·2001 · AMC 12A·Hard

In rectangle

ABCD

, points

F

and

G

lie on

AB

so that

AF=FG=GB

and

E

is the midpoint of

\overline{DC}

. Also,

\overline{AC}

intersects

\overline{EF}

at

H

and

\overline{EG}

at

J

. The area of the rectangle

ABCD

is

70

. Find the area of triangle

EHJ

.

AMC 12·2001 · AMC 12A·Hard

In

\triangle ABC

,

\angle ABC=45^\circ

. Point

D

is on

\overline{BC}

so that

2\cdot BD=CD

and

\angle DAB=15^\circ

. Find

\angle ACB

.

AMC 12·2001 · AMC 12A·Easy

Which of the cones listed below can be formed from a

252^\circ

sector of a circle of radius

10

by aligning the two straight sides?

AMC 12·2002 · AMC 12A·Medium

Let

C_1

and

C_2

be circles defined by

(x-10)^2 + y^2 = 36

and

(x+15)^2 + y^2 = 81

respectively. What is the length of the shortest line segment

PQ

that is tangent to

C_1

at

P

and to

C_2

at

Q

?

AMC 12·2002 · AMC 12A·Hard

In triangle

ABC

, side

AC

and the perpendicular bisector of

BC

meet in point

D

, and

BD

bisects

\angle ABC

. If

AD=9

and

DC=7

, what is the area of triangle

ABD

?

AMC 12·2002 · AMC 12A·Easy

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

AMC 12·2002 · AMC 12A·Easy

A

45^\circ

arc of circle A is equal in length to a

30^\circ

arc of circle B. What is the ratio of circle A's area and circle B's area?

AMC 12·2002 · AMC 12A·Easy

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let

B

be the total area of the blue triangles,

W

the total area of the white squares, and

R

the area of the red square. Which of the following is correct?

AMC 12·2002 · AMC 12B·Medium

Let

\triangle XOY

be a right-angled triangle with

\angle XOY = 90^{\circ}

. Let

M

and

N

be the midpoints of legs

OX

and

OY

, respectively. Given that

XN = 19

and

YM = 22

, find

XY

.

AMC 12·2002 · AMC 12B·Hard

In

\triangle ABC

, we have

AB = 1

and

AC = 2

. Side

\overline{BC}

and the median from

A

to

\overline{BC}

have the same length. What is

BC

?

AMC 12·2002 · AMC 12B·Hard

A convex quadrilateral

ABCD

with area

2002

contains a point

P

in its interior such that

PA = 24, PB = 32, PC = 28, PD = 45

. Find the perimeter of

ABCD

.

AMC 12·2002 · AMC 12B·Easy

Let

v, w, x, y,

and

z

be the degree measures of the five angles of a pentagon. Suppose that

v < w < x < y < z

and

v, w, x, y,

and

z

form an arithmetic sequence. Find the value of

x

.

AMC 12·2002 · AMC 12 Practice·Medium

The altitudes of a triangle are

12, 15,

and

20.

The largest angle in this triangle is

AMC 12·2002 · AMC 12 Practice·Medium

In quadrilateral

ABCD

,

m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4,

and

CD=5.

Find the area of

ABCD.

AMC 12·2002 · AMC 12 Practice·Easy

Let

AB

be a segment of length

26

, and let points

C

and

D

be located on

AB

such that

AC=1

and

AD=8

. Let

E

and

F

be points on one of the semicircles with diameter

AB

for which

EC

and

FD

are perpendicular to

AB

. Find

EF.

AMC 12·2003 · AMC 12A·Medium

A square and an equilateral triangle have the same perimeter. Let

A

be the area of the circle circumscribed about the square and

B

the area of the circle circumscribed around the triangle. Find

A/B

.

AMC 12·2003 · AMC 12A·Medium

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

AMC 12·2003 · AMC 12A·Medium

A semicircle of diameter

1

sits at the top of a semicircle of diameter

2

, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a ''lune''. Determine the area of this lune.

AMC 12·2003 · AMC 12A·Medium

Square

ABCD

has sides of length

4

, and

M

is the midpoint of

\overline{CD}

. A circle with radius

2

and center

M

intersects a circle with radius

4

and center

A

at points

P

and

D

. What is the distance from

P

to

\overline{AD}

?

AMC 12·2003 · AMC 12A·Easy

A solid box is

15

cm by

10

cm by

8

cm. A new solid is formed by removing a cube

3

cm on a side from each corner of this box. What percent of the original volume is removed?

AMC 12·2003 · AMC 12A·Easy

How many non-congruent triangles with perimeter

7

have integer side lengths?

AMC 12·2003 · AMC 12B·Medium

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies

75\%

of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius ?

AMC 12·2003 · AMC 12B·Medium

Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?

AMC 12·2003 · AMC 12B·Hard

Let

ABCD

be a rhombus with

AC = 16

and

BD = 30

. Let

N

be a point on

\overline{AB}

, and let

P

and

Q

be the feet of the perpendiculars from

N

to

\overline{AC}

and

\overline{BD}

, respectively. Which of the following is closest to the minimum possible value of

PQ

?

AMC 12·2004 · AMC 12A·Medium

Circles

A, B

and

C

are externally tangent to each other, and internally tangent to circle

D

. Circles

B

and

C

are congruent. Circle

A

has radius

1

and passes through the center of

D

. What is the radius of circle

B

?

AMC 12·2004 · AMC 12A·Easy

In the overlapping triangles

\triangle{ABC}

and

\triangle{ABE}

sharing common side

AB

,

\angle{EAB}

and

\angle{ABC}

are right angles,

AB=4

,

BC=6

,

AE=8

, and

\overline{AC}

and

\overline{BE}

intersect at

D

. What is the difference between the areas of

\triangle{ADE}

and

\triangle{BDC}

?

AMC 12·2004 · AMC 12B·Easy

An annulus is the region between two concentric circles. The concentric circles in the ﬁgure have radii

b

and

c

, with

b>c

. Let

OX

be a radius of the larger circle, let

XZ

be tangent to the smaller circle at

Z

, and let

OY

be the radius of the larger circle that contains

Z

. Let

a=XZ

,

d=YZ

, and

e=XY

. What is the area of the annulus?

AMC 12·2004 · AMC 12B·Medium

In

\triangle ABC

,

AB=13

,

AC=5

, and

BC=12

. Points

M

and

N

lie on

AC

and

BC

, respectively, with

CM=CN=4

. Points

J

and

K

are on

AB

so that

MJ

and

NK

are perpendicular to

AB

. What is the area of pentagon

CMJKN

?

AMC 12·2004 · AMC 12B·Medium

A truncated cone has horizontal bases with radii

18

and

2

. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?

AMC 12·2004 · AMC 12B·Hard

In

\triangle ABC

,

AB = BC

, and

\overline{BD}

is an altitude. Point

E

is on the extension of

\overline{AC}

such that

BE = 10

. The values of

\tan \angle CBE

,

\tan \angle DBE

, and

\tan \angle ABE

form a geometric progression, and the values of

\cot \angle DBE, \cot \angle CBE, \cot \angle DBC

form an arithmetic progression. What is the area of

\triangle ABC

?

AMC 12·2004 · AMC 12B·Easy

The point

(-3,2)

is rotated

90^\circ

clockwise around the origin to point

B

. Point

B

is then reflected over the line

x=y

to point

C

. What are the coordinates of

C

?

AMC 12·2005 · AMC 12A·Easy

A wooden cube

n

units on a side is painted red on all six faces and then cut into

n^3

unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is

n

?

AMC 12·2005 · AMC 12A·Medium

Let

\overline{AB}

be a diameter of a circle and

C

be a point on

\overline{AB}

with

2 \cdot AC = BC

. Let

D

and

E

be points on the circle such that

\overline{DC} \perp \overline{AB}

and

\overline{DE}

is a second diameter. What is the ratio of the area of

\triangle DCE

to the area of

\triangle ABD

?

AMC 12·2005 · AMC 12A·Medium

Three circles of radius

s

are drawn in the first quadrant of the

xy

-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the

x

-axis, and the third is tangent to the first circle and the

y

-axis. A circle of radius

r > s

is tangent to both axes and to the second and third circles. What is

r/s

?

AMC 12·2005 · AMC 12A·Medium

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure

1

. The cube is then cut in the same manner along the dashed lines shown in Figure

2

. This creates nine pieces. What is the volume of the piece that contains vertex

W

?

AMC 12·2005 · AMC 12A·Hard

A rectangular box

P

is inscribed in a sphere of radius

r

. The surface area of

P

is 384, and the sum of the lengths of its 12 edges is 112. What is

r

?

AMC 12·2005 · AMC 12A·Hard

Let

S

be the set of all points with coordinates

(x,y,z)

, where

x

,

y

, and

z

are each chosen from the set

\{0, 1, 2\}

. How many equilateral triangles have all their vertices in

S

?

AMC 12·2005 · AMC 12A·Easy

A rectangle with a diagonal of length

x

is twice as long as it is wide. What is the area of the rectangle?

AMC 12·2005 · AMC 12A·Easy

Square

EFGH

is inside the square

ABCD

so that each side of

EFGH

can be extended to pass through a vertex of

ABCD

. Square

ABCD

has side length

\sqrt{50}

,

E

is between

B

and

H

, and

BE = 1

. What is the area of the inner square

EFGH

?

AMC 12·2005 · AMC 12B·Medium

A circle having center

(0,k)

, with

k>6

, is tangent to the lines

y=x

,

y=-x

and

y=6

. What is the radius of this circle?

AMC 12·2005 · AMC 12B·Medium

Let

A(2,2)

and

B(7,7)

be points in the plane. Define

R

as the region in the first quadrant consisting of those points

C

such that

\triangle ABC

is an acute triangle. What is the closest integer to the area of the region

R

?

AMC 12·2005 · AMC 12B·Hard

All three vertices of an equilateral triangle are on the parabola

y = x^2

, and one of its sides has a slope of

2

. The

x

-coordinates of the three vertices have a sum of

m/n

, where

m

and

n

are relatively prime positive integers. What is the value of

m + n

?

AMC 12·2005 · AMC 12B·Easy

An

8

-foot by

10

-foot floor is tiled with square tiles of size

1

foot by

1

foot. Each tile has a pattern consisting of four white quarter circles of radius

1/2

foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

AMC 12·2005 · AMC 12B·Easy

In

\triangle ABC

, we have

AC=BC=7

and

AB=2

. Suppose that

D

is a point on line

AB

such that

B

lies between

A

and

D

and

CD=8

. What is

BD

?

AMC 12·2005 · AMC 12B·Easy

What is the area enclosed by the graph of

|3x|+|4y|=12

?

AMC 12·2006 · AMC 12A·Medium

A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

AMC 12·2006 · AMC 12A·Medium

The vertices of a

3-4-5

right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?

AMC 12·2006 · AMC 12A·Medium

Circles with centers

A

and

B

have radii

3

and

8

, respectively. A common internal tangent intersects the circles at

C

and

D

, respectively. Lines

AB

and

CD

intersect at

E

, and

AE=5

. What is

CD

?

AMC 12·2006 · AMC 12A·Medium

Square

ABCD

has side length

s

, a circle centered at

E

has radius

r

, and

r

and

s

are both rational. The circle passes through

D

, and

D

lies on

\overline{BE}

. Point

F

lies on the circle, on the same side of

\overline{BE}

as

A

. Segment

AF

is tangent to the circle, and

AF=\sqrt{9+5\sqrt{2}}

. What is

r/s

?

AMC 12·2006 · AMC 12A·Medium

Circles with centers

(2,4)

and

(14,9)

have radii

4

and

9

, respectively. The equation of a common external tangent to the circles can be written in the form

y=mx+b

with

m>0

. What is

b

?

AMC 12·2006 · AMC 12A·Easy

The

8\times 18

rectangle

ABCD

is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is

y

?

AMC 12·2006 · AMC 12B·Medium

Circles with centers

O

and

P

have radii 2 and 4, respectively, and are externally tangent. Points

A

and

B

are on the circle centered at

O

, and points

C

and

D

are on the circle centered at

P

, such that

\overline{AD}

and

\overline{BC}

are common external tangents to the circles. What is the area of hexagon

AOBCPD

?

AMC 12·2006 · AMC 12B·Medium

Regular hexagon

ABCDEF

has vertices

A

and

C

at

(0,0)

and

(7,1)

, respectively. What is its area?

AMC 12·2006 · AMC 12B·Hard

Rectangle

ABCD

has area

2006

. An ellipse with area

2006\pi

passes through

A

and

C

and has foci at

B

and

D

. What is the perimeter of the rectangle? (The area of an ellipse is

ab\pi

where

2a

and

2b

are the lengths of the axes.)

AMC 12·2006 · AMC 12B·Hard

Isosceles

\triangle ABC

has a right angle at

C

. Point

P

is inside

\triangle ABC

, such that

PA=11

,

PB=7

, and

PC=6

. Legs

\overline{AC}

and

\overline{BC}

have length

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

, where

a

and

b

are positive integers. What is

a+b

?

AMC 12·2006 · AMC 12B·Hard

Let

S

be the set of all point

(x,y)

in the coordinate plane such that

0 \le x \le \frac{\pi}{2}

and

0 \le y \le \frac{\pi}{2}

. What is the area of the subset of

S

for which

\sin^2x-\sin x \sin y + \sin^2y \le \frac34?

AMC 12·2007 · AMC 12A·Easy

A triangle with side lengths in the ratio

3 : 4 : 5

is inscribed in a circle with radius 3. What is the area of the triangle?

AMC 12·2007 · AMC 12A·Medium

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?

AMC 12·2007 · AMC 12A·Easy

Triangles

ABC

and

ADC

are isosceles with

AB=BC

and

AD=DC

. Point

D

is inside triangle

ABC

, angle

ABC

measures 40 degrees, and angle

ADC

measures 140 degrees. What is the degree measure of angle

BAD

?

AMC 12·2007 · AMC 12A·Easy

A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?

AMC 12·2008 · AMC 12A·Medium

Points

A

and

B

lie on a circle centered at

O

, and

\angle AOB = 60^\circ

. A second circle is internally tangent to the first and tangent to both

\overline{OA}

and

\overline{OB}

. What is the ratio of the area of the smaller circle to that of the larger circle?

AMC 12·2008 · AMC 12A·Medium

What is the area of the region defined by the inequality

|3x-18|+|2y+7|\le3

?

AMC 12·2008 · AMC 12A·Medium

Triangle

ABC

, with sides of length

5

,

6

, and

7

, has one vertex on the positive

x

-axis, one on the positive

y

-axis, and one on the positive

z

-axis. Let

O

be the origin . What is the volume of tetrahedron

OABC

?

AMC 12·2008 · AMC 12A·Medium

Triangle

ABC

has

AC=3

,

BC=4

, and

AB=5

. Point

D

is on

\overline{AB}

, and

\overline{CD}

bisects the right angle. The inscribed circles of

\triangle ADC

and

\triangle BCD

have radii

r_a

and

r_b

, respectively. What is

r_a/r_b

?

AMC 12·2008 · AMC 12A·Hard

A round table has radius

4

. Six rectangular place mats are placed on the table. Each place mat has width

1

and length

x

as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length

x

. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is

x

?

AMC 12·2008 · AMC 12A·Hard

The solutions of the equation

z^4+4z^3i-6z^2-4zi-i=0

are the vertices of a convex polygon in the complex plane. What is the area of the polygon?

AMC 12·2008 · AMC 12A·Hard

Triangle

ABC

has

\angle C = 60^{\circ}

and

BC = 4

. Point

D

is the midpoint of

BC

. What is the largest possible value of

\tan{\angle BAD}

?

AMC 12·2008 · AMC 12A·Easy

What is the volume of a cube whose surface area is twice that of a cube with volume 1?

AMC 12·2008 · AMC 12B·Medium

A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top

\frac{1}{8}

of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?

AMC 12·2008 · AMC 12B·Medium

Vertex

E

of equilateral triangle

\triangle ABE

is in the interior of unit square

ABCD

. Let

R

be the region consisting of all points inside

ABCD

and outside

\triangle ABE

whose distance from

\overline{AD}

is between

\frac{1}{3}

and

\frac{2}{3}

. What is the area of

R

? (Solution)

AMC 12·2008 · AMC 12B·Medium

A circle has a radius of

\log_{10}{(a^2)}

and a circumference of

\log_{10}{(b^4)}

. What is

\log_{a}{b}

? (Solution)

AMC 12·2008 · AMC 12B·Medium

On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let

R

be the region formed by the union of the square and all the triangles, and

S

be the smallest convex polygon that contains

R

. What is the area of the region that is inside

S

but outside

R

?

AMC 12·2008 · AMC 12B·Medium

A rectangular floor measures

a

by

b

feet, where

a

and

b

are positive integers with

b > a

. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width

1

foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair

(a,b)

? (Solution)

AMC 12·2008 · AMC 12B·Medium

Let

A

,

B

and

C

be three distinct points on the graph of

y=x^2

such that line

AB

is parallel to the

x

-axis and

\triangle ABC

is a right triangle with area

2008

. What is the sum of the digits of the

y

-coordinate of

C

? (Solution)

AMC 12·2008 · AMC 12B·Medium

A pyramid has a square base

ABCD

and vertex

E

. The area of square

ABCD

is

196

, and the areas of

\triangle ABE

and

\triangle CDE

are

105

and

91

, respectively. What is the volume of the pyramid?

AMC 12·2008 · AMC 12B·Hard

Let

A_0 = (0,0)

. Distinct points

A_1,A_2,\ldots

lie on the

x

-axis, and distinct points

B_1,B_2,\ldots

lie on the graph of

y = \sqrt {x}

. For every positive integer

n

,

A_{n - 1}B_nA_n

is an equilateral triangle. What is the least

n

for which the length

A_0A_n\ge100

? (Solution)

AMC 12·2008 · AMC 12B·Hard

Let

ABCD

be a trapezoid with

AB||CD, AB=11, BC=5, CD=19,

and

DA=7

. Bisectors of

\angle A

and

\angle D

meet at

P

, and bisectors of

\angle B

and

\angle C

meet at

Q

. What is the area of hexagon

ABQCDP

?

AMC 12·2008 · AMC 12B·Easy

On circle

O

, points

C

and

D

are on the same side of diameter

\overline{AB}

,

\angle AOC = 30^\circ

, and

\angle DOB = 45^\circ

. What is the ratio of the area of the smaller sector

COD

to the area of the circle? (Solution)

AMC 12·2008 · AMC 12B·Easy

Points

A

and

B

are on a circle of radius

5

and

AB = 6

. Point

C

is the midpoint of the minor arc

AB

. What is the length of the line segment

AC

?

AMC 12·2009 · AMC 12A·Easy

In quadrilateral

ABCD

,

AB = 5

,

BC = 17

,

CD = 5

,

DA = 9

, and

BD

is an integer. What is

BD

?

AMC 12·2009 · AMC 12A·Medium

A triangle has vertices

(0,0)

,

(1,1)

, and

(6m,0)

, and the line

y = mx

divides the triangle into two triangles of equal area. What is the sum of all possible values of

m

?

AMC 12·2009 · AMC 12A·Medium

A circle with center

C

is tangent to the positive

x

and

y

-axes and externally tangent to the circle centered at

(3,0)

with radius

1

. What is the sum of all possible radii of the circle with center

C

?

AMC 12·2009 · AMC 12A·Medium

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were

A

and

B

, respectively. Each polygon had a side length of

2

. Which of the following is true?

AMC 12·2009 · AMC 12A·Medium

Convex quadrilateral

ABCD

has

AB = 9

and

CD = 12

. Diagonals

AC

and

BD

intersect at

E

,

AC = 14

, and

\triangle AED

and

\triangle BEC

have equal areas. What is

AE

?

AMC 12·2009 · AMC 12A·Hard

A regular octahedron has side length

1

. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area

\frac {a\sqrt {b}}{c}

, where

a

,

b

, and

c

are positive integers,

a

and

c

are relatively prime, and

b

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

a + b + c

?

AMC 12·2009 · AMC 12A·Easy

One dimension of a cube is increased by

1

, another is decreased by

1

, and the third is left unchanged. The volume of the new rectangular solid is

5

less than that of the cube. What was the volume of the cube?

AMC 12·2009 · AMC 12A·Easy

Four congruent rectangles are placed as shown. The area of the outer square is

4

times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

AMC 12·2009 · AMC 12B·Medium

Triangle

ABC

has

AB = 13

and

AC = 15

, and the altitude to

\overline{BC}

has length

12

. What is the sum of the two possible values of

BC

?

AMC 12·2009 · AMC 12B·Medium

Trapezoid

ABCD

has

AD||BC

,

BD = 1

,

\angle DBA = 23^{\circ}

, and

\angle BDC = 46^{\circ}

. The ratio

BC: AD

is

9: 5

. What is

CD

?

AMC 12·2009 · AMC 12B·Medium

A convex polyhedron

Q

has vertices

V_1,V_2,\ldots,V_n

, and

100

edges. The polyhedron is cut by planes

P_1,P_2,\ldots,P_n

in such a way that plane

P_k

cuts only those edges that meet at vertex

V_k

. In addition, no two planes intersect inside or on

Q

. The cuts produce

n

pyramids and a new polyhedron

R

. How many edges does

R

have?

AMC 12·2009 · AMC 12B·Hard

Parallelogram

ABCD

has area

1,\!000,\!000

. Vertex

A

is at

(0,0)

and all other vertices are in the first quadrant. Vertices

B

and

D

are lattice points on the lines

y = x

and

y = kx

for some integer

k > 1

, respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.)

AMC 12·2009 · AMC 12B·Hard

The set

G

is defined by the points

(x,y)

with integer coordinates,

3\le|x|\le7

,

3\le|y|\le7

. How many squares of side length at least

6

have their four vertices in

G

?

AMC 12·2009 · AMC 12B·Easy

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths

15

and

25

meters. What fraction of the yard is occupied by the flower beds?

AMC 12·2009 · AMC 12B·Easy

Triangle

ABC

has vertices

A = (3,0)

,

B = (0,3)

, and

C

, where

C

is on the line

x + y = 7

. What is the area of

\triangle ABC

?

AMC 12·2010 · AMC 12A·Medium

Nondegenerate

\triangle ABC

has integer side lengths,

\overline{BD}

is an angle bisector,

AD = 3

, and

DC=8

. What is the smallest possible value of the perimeter?

AMC 12·2010 · AMC 12A·Medium

Equiangular hexagon

ABCDEF

has side lengths

AB=CD=EF=1

and

BC=DE=FA=r

. The area of

\triangle ACE

is

70\%

of the area of the hexagon. What is the sum of all possible values of

r

?

AMC 12·2010 · AMC 12A·Hard

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?

AMC 12·2010 · AMC 12A·Easy

Rectangle

ABCD

, pictured below, shares

50\%

of its area with square

EFGH

. Square

EFGH

shares

20\%

of its area with rectangle

ABCD

. What is

\frac{AB}{AD}

?

AMC 12·2010 · AMC 12A·Easy

Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?

AMC 12·2010 · AMC 12A·Easy

Triangle

ABC

has

AB=2 \cdot AC

. Let

D

and

E

be on

\overline{AB}

and

\overline{BC}

, respectively, such that

\angle BAE = \angle ACD

. Let

F

be the intersection of segments

AE

and

CD

, and suppose that

\triangle CFE

is equilateral. What is

\angle ACB

?

AMC 12·2010 · AMC 12A·Easy

A solid cube has side length

3

inches. A

2

-inch by

2

-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

AMC 12·2010 · AMC 12B·Medium

In

\triangle ABC

,

\cos(2A-B)+\sin(A+B)=2

and

AB=4

. What is

BC

?

AMC 12·2010 · AMC 12B·Hard

Let

ABCD

be a cyclic quadrilateral. The side lengths of

ABCD

are distinct integers less than

15

such that

BC\cdot CD=AB\cdot DA

. What is the largest possible value of

BD

?

AMC 12·2011 · AMC 12A·Medium

Circles

A, B,

and

C

each have radius 1. Circles

A

and

B

share one point of tangency. Circle

C

has a point of tangency with the midpoint of

\overline{AB}.

What is the area inside circle

C

but outside circle

A

and circle

B?

AMC 12·2011 · AMC 12A·Medium

Triangle

ABC

has side-lengths

AB = 12, BC = 24,

and

AC = 18.

The line through the incenter of

\triangle ABC

parallel to

\overline{BC}

intersects

\overline{AB}

at

M

and

\overline{AC}

at

N.

What is the perimeter of

\triangle AMN?

AMC 12·2011 · AMC 12A·Medium

The circular base of a hemisphere of radius

2

rests on the base of a square pyramid of height

6

. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?

AMC 12·2011 · AMC 12A·Medium

Circles with radii

1

,

2

, and

3

are mutually externally tangent. What is the area of the triangle determined by the points of tangency?

AMC 12·2011 · AMC 12A·Easy

There are

5

coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?

AMC 12·2011 · AMC 12A·Hard

Let

R

be a unit square region and

n \geq 4

an integer. A point

X

in the interior of

R

is called ''n-ray partitional'' if there are

n

rays emanating from

X

that divide

R

into

n

triangles of equal area. How many points are

100

-ray partitional but not

60

-ray partitional?

AMC 12·2011 · AMC 12A·Hard

Consider all quadrilaterals

ABCD

such that

AB=14

,

BC=9

,

CD=7

, and

DA=12

. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?

AMC 12·2011 · AMC 12A·Hard

Triangle

ABC

has

\angle BAC = 60^{\circ}

,

\angle CBA \leq 90^{\circ}

,

BC=1

, and

AC \geq AB

. Let

H

,

I

, and

O

be the orthocenter, incenter, and circumcenter of

\triangle ABC

, respectively. Assume that the area of pentagon

BCOIH

is the maximum possible. What is

\angle CBA

?

AMC 12·2011 · AMC 12B·Easy

Rectangle

ABCD

has

AB=6

and

BC=3

. Point

M

is chosen on side

AB

so that

\angle AMD=\angle CMD

. What is the degree measure of

\angle AMD

?

AMC 12·2011 · AMC 12B·Medium

A frog located at

(x,y)

, with both

x

and

y

integers, makes successive jumps of length

5

and always lands on points with integer coordinates. Suppose that the frog starts at

(0,0)

and ends at

(1,0)

. What is the smallest possible number of jumps the frog makes?

AMC 12·2011 · AMC 12B·Medium

A segment through the focus

F

of a parabola with vertex

V

is perpendicular to

\overline{FV}

and intersects the parabola in points

A

and

B

. What is

\cos\left(\angle AVB\right)

?

AMC 12·2011 · AMC 12B·Medium

Rhombus

ABCD

has side length

2

and

\angle B = 120^{\circ}

. Region

R

consists of all points inside of the rhombus that are closer to vertex

B

than any of the other three vertices. What is the area of

R

?

AMC 12·2011 · AMC 12B·Medium

A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?

AMC 12·2011 · AMC 12B·Medium

A lattice point in an

xy

-coordinate system is any point

(x, y)

where both

x

and

y

are integers. The graph of

y = mx + 2

passes through no lattice point with

0 < x \leq 100

for all

m

such that

\frac{1}{2} < m < a

. What is the maximum possible value of

a

?

AMC 12·2011 · AMC 12B·Medium

Triangle

ABC

has

AB = 13, BC = 14

, and

AC = 15

. The points

D, E

, and

F

are the midpoints of

\overline{AB}, \overline{BC}

, and

\overline{AC}

respectively. Let

X \not= E

be the intersection of the circumcircles of

\Delta BDE

and

\Delta CEF

. What is

XA + XB + XC

?

AMC 12·2011 · AMC 12B·Hard

Let

T_1

be a triangle with side lengths

2011, 2012

, and

2013

. For

n \geq 1

, if

T_n = \Delta ABC

and

D, E

, and

F

are the points of tangency of the incircle of

\Delta ABC

to the sides

AB, BC

, and

AC

, respectively, then

T_{n+1}

is a triangle with side lengths

AD, BE

, and

CF

, if it exists. What is the perimeter of the last triangle in the sequence

\left(T_n\right)

?

AMC 12·2011 · AMC 12B·Easy

Two tangents to a circle are drawn from a point

A

. The points of contact

B

and

C

divide the circle into arcs with lengths in the ratio

2 : 3

. What is the degree measure of

\angle{BAC}

?

AMC 12·2011 · AMC 12B·Easy

Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of

6

meters, and it takes her

36

seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?

AMC 12·2012 · AMC 12A·Easy

A triangle has area

30

, one side of length

10

, and the median to that side of length

9

. Let

\theta

be the acute angle formed by that side and the median. What is

\sin{\theta}

?

AMC 12·2012 · AMC 12A·Medium

A square region

ABCD

is externally tangent to the circle with equation

x^2+y^2=1

at the point

(0,1)

on the side

CD

. Vertices

A

and

B

are on the circle with equation

x^2+y^2=4

. What is the side length of this square?

AMC 12·2012 · AMC 12A·Medium

The closed curve in the figure is made up of

9

congruent circular arcs each of length

\frac{2\pi}{3}

, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side

2

. What is the area enclosed by the curve?

AMC 12·2012 · AMC 12A·Medium

Circle

C_1

has its center

O

lying on circle

C_2

. The two circles meet at

X

and

Y

. Point

Z

in the exterior of

C_1

lies on circle

C_2

and

XZ=13

,

OZ=11

, and

YZ=7

. What is the radius of circle

C_1

?

AMC 12·2012 · AMC 12A·Medium

Triangle

ABC

has

AB=27

,

AC=26

, and

BC=25

. Let

I

be the intersection of the internal angle bisectors of

\triangle ABC

. What is

BI

?

AMC 12·2012 · AMC 12A·Hard

Distinct planes

p_1,p_2,....,p_k

intersect the interior of a cube

Q

. Let

S

be the union of the faces of

Q

and let

P =\bigcup_{j=1}^{k}p_{j}

. The intersection of

P

and

S

consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of

Q

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

k

?

AMC 12·2012 · AMC 12B·Easy

What is the area of the polygon whose vertices are the points of intersection of the curves

x^2 + y^2 =25

and

(x-4)^2 + 9y^2 = 81

?

AMC 12·2012 · AMC 12B·Medium

Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral surface of each cone. What is the ratio of the volume of the smaller cone to the larger cone?

AMC 12·2012 · AMC 12B·Medium

Square

PQRS

lies in the first quadrant. Points

(3,0), (5,0), (7,0),

and

(13,0)

lie on lines

SP, RQ, PQ

, and

SR

, respectively. What is the sum of the coordinates of the center of the square

PQRS

?

AMC 12·2012 · AMC 12B·Medium

A unit cube has vertices

P_1,P_2,P_3,P_4,P_1',P_2',P_3',

and

P_4'

. Vertices

P_2

,

P_3

, and

P_4

are adjacent to

P_1

, and for

1\le i\le 4,

vertices

P_i

and

P_i'

are opposite to each other. A regular octahedron has one vertex in each of the segments

P_1P_2

,

P_1P_3

,

P_1P_4

,

P_1'P_2'

,

P_1'P_3'

, and

P_1'P_4'

. What is the octahedron's side length?

AMC 12·2012 · AMC 12B·Easy

A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?

AMC 12·2012 · AMC 12B·Medium

A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of

r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3

, where

r_1

,

r_2

, and

r_3

are rational numbers and

n_1

and

n_2

are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to

r_1+r_2+r_3+n_1+n_2

?

AMC 12·2012 · AMC 12B·Hard

Square

AXYZ

is inscribed in equiangular hexagon

ABCDEF

with

X

on

\overline{BC}

,

Y

on

\overline{DE}

, and

Z

on

\overline{EF}

. Suppose that

AB=40

, and

EF=41(\sqrt{3}-1)

. What is the side-length of the square?

AMC 12·2012 · AMC 12B·Hard

Let

S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}

. Let

T

be the set of all right triangles whose vertices are in

S

. For every right triangle

t=\triangle{ABC}

with vertices

A

,

B

, and

C

in counter-clockwise order and right angle at

A

, let

f(t)=\tan(\angle{CBA})

. What is

\prod_{t\in T} f(t)?

AMC 12·2013 · AMC 12A·Easy

Square

ABCD

has side length

10

. Point

E

is on

\overline{BC}

, and the area of

\bigtriangleup ABE

is

40

. What is

BE

?

AMC 12·2013 · AMC 12A·Medium

Triangle

ABC

is equilateral with

AB=1

. Points

E

and

G

are on

\overline{AC}

and points

D

and

F

are on

\overline{AB}

such that both

\overline{DE}

and

\overline{FG}

are parallel to

\overline{BC}

. Furthermore, triangle

ADE

and trapezoids

DFGE

and

FBCG

all have the same perimeter. What is

DE+FG

?

AMC 12·2013 · AMC 12A·Medium

The angles in a particular triangle are in arithmetic progression, and the side lengths are

4,5,x

. The sum of the possible values of

x

equals

a+\sqrt{b}+\sqrt{c}

where

a, b

, and

c

are positive integers. What is

a+b+c

?

AMC 12·2013 · AMC 12A·Medium

Let points

A = (0,0) , \ B = (1,2), \ C = (3,3),

and

D = (4,0)

. Quadrilateral

ABCD

is cut into equal area pieces by a line passing through

A

. This line intersects

\overline{CD}

at point

\left (\frac{p}{q}, \frac{r}{s} \right )

, where these fractions are in lowest terms. What is

p + q + r + s

?

AMC 12·2013 · AMC 12A·Medium

Six spheres of radius

1

are positioned so that their centers are at the vertices of a regular hexagon of side length

2

. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?

AMC 12·2013 · AMC 12A·Medium

In

\bigtriangleup ABC

,

AB = 86

, and

AC = 97

. A circle with center

A

and radius

AB

intersects

\overline{BC}

at points

B

and

X

. Moreover

\overline{BX}

and

\overline{CX}

have integer lengths. What is

BC

?

AMC 12·2013 · AMC 12A·Hard

ABCD

is a square of side length

\sqrt{3} + 1

. Point

P

is on

\overline{AC}

such that

AP = \sqrt{2}

. The square region bounded by

ABCD

is rotated

90^{\circ}

counterclockwise with center

P

, sweeping out a region whose area is

\frac{1}{c} (a \pi + b)

, where

a

,

b

, and

c

are positive integers and

\gcd(a,b,c) = 1

. What is

a + b + c

?

AMC 12·2013 · AMC 12A·Easy

In

\triangle ABC

,

AB=AC=28

and

BC=20

. Points

D,E,

and

F

are on sides

\overline{AB}

,

\overline{BC}

, and

\overline{AC}

, respectively, such that

\overline{DE}

and

\overline{EF}

are parallel to

\overline{AC}

and

\overline{AB}

, respectively. What is the perimeter of parallelogram

ADEF

?

AMC 12·2013 · AMC 12B·Medium

The internal angles of quadrilateral

ABCD

form an arithmetic progression. Triangles

ABD

and

DCB

are similar with

\angle DBA = \angle DCB

and

\angle ADB = \angle CBD

. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of

ABCD

?

AMC 12·2013 · AMC 12B·Medium

Let

ABCDE

be an equiangular convex pentagon of perimeter

1

. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let

s

be the perimeter of this star. What is the difference between the maximum and the minimum possible values of

s

?

AMC 12·2013 · AMC 12B·Medium

In triangle

ABC

,

AB=13

,

BC=14

, and

CA=15

. Distinct points

D

,

E

, and

F

lie on segments

\overline{BC}

,

\overline{CA}

, and

\overline{DE}

, respectively, such that

\overline{AD}\perp\overline{BC}

,

\overline{DE}\perp\overline{AC}

, and

\overline{AF}\perp\overline{BF}

. The length of segment

\overline{DF}

can be written as

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

?

AMC 12·2013 · AMC 12B·Medium

For

135^\circ < x < 180^\circ

, points

P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)

and

S =(\tan x, \tan^2 x)

are the vertices of a trapezoid. What is

\sin(2x)

?

AMC 12·2013 · AMC 12B·Hard

Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form

y=ax+b

with

a

and

b

integers such that

a\in \{-2,-1,0,1,2\}

and

b\in \{-3,-2,-1,1,2,3\}

. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?

AMC 12·2013 · AMC 12B·Hard

Let

ABC

be a triangle where

M

is the midpoint of

\overline{AC}

, and

\overline{CN}

is the angle bisector of

\angle{ACB}

with

N

on

\overline{AB}

. Let

X

be the intersection of the median

\overline{BM}

and the bisector

\overline{CN}

. In addition

\triangle BXN

is equilateral with

AC=2

. What is

BX^2

?

AMC 12·2013 · AMC 12B·Easy

Line

l_1

has equation

3x - 2y = 1

and goes through

A = (-1, -2)

. Line

l_2

has equation

y = 1

and meets line

l_1

at point

B

. Line

l_3

has positive slope, goes through point

A

, and meets

l_2

at point

C

. The area of

\triangle ABC

is

3

. What is the slope of

l_3

?

AMC 12·2014 · AMC 12A·Easy

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length

1

. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

AMC 12·2014 · AMC 12A·Medium

Two circles intersect at points

A

and

B

. The minor arcs

AB

measure

30^\circ

on one circle and

60^\circ

on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?

AMC 12·2014 · AMC 12A·Medium

A

4\times 4\times h

rectangular box contains a sphere of radius

2

and eight smaller spheres of radius

1

. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is

h

?

AMC 12·2014 · AMC 12A·Medium

In

\triangle BAC

,

\angle BAC=40^\circ

,

AB=10

, and

AC=6

. Points

D

and

E

lie on

\overline{AB}

and

\overline{AC}

respectively. What is the minimum possible value of

BE+DE+CD

?

AMC 12·2014 · AMC 12B·Medium

Real numbers

a

and

b

are chosen with

1<a<b

such that no triangle with positive area has side lengths

1

,

a

, and

b

or

\frac{1}{b}

,

\frac{1}{a}

, and

1

. What is the smallest possible value of

b

?

AMC 12·2014 · AMC 12B·Medium

A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?

AMC 12·2014 · AMC 12B·Medium

The numbers

1

,

2

,

3

,

4

,

5

, are to be arranged in a circle. An arrangement is

\textit{bad}

if it is not true that for every

n

from

1

to

15

one can find a subset of the numbers that appear consecutively on the circle that sum to

n

. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

AMC 12·2014 · AMC 12B·Medium

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?

AMC 12·2014 · AMC 12B·Hard

In the figure,

ABCD

is a square of side length

1

. The rectangles

JKHG

and

EBCF

are congruent. What is

BE

?

AMC 12·2014 · AMC 12B·Hard

Let

ABCDE

be a pentagon inscribed in a circle such that

AB = CD = 3

,

BC = DE = 10

, and

AE= 14

. The sum of the lengths of all diagonals of

ABCDE

is equal to

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

?

AMC 12·2014 · AMC 12B·Easy

Doug constructs a square window using

8

equal-size panes of glass, as shown. The ratio of the height to width for each pane is

5 : 2

, and the borders around and between the panes are

2

inches wide. In inches, what is the side length of the square window?

AMC 12·2014 · AMC 12B·Easy

Convex quadrilateral

ABCD

has

AB=3

,

BC=4

,

CD=13

,

AD=12

, and

\angle ABC=90^{\circ}

, as shown. What is the area of the quadrilateral?

AMC 12·2015 · AMC 12A·Medium

Tetrahedron

ABCD

has

AB=5,AC=3,BC=4,BD=4,AD=3,

and

CD=\frac{12}{5}\sqrt{2}

. What is the volume of the tetrahedron?

AMC 12·2015 · AMC 12A·Medium

For some positive integers

p

, there is a quadrilateral

ABCD

with positive integer side lengths, perimeter

p

, right angles at

B

and

C

,

AB=2

, and

CD=AD

. How many different values of

p<2015

are possible?

AMC 12·2015 · AMC 12A·Easy

Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?

AMC 12·2015 · AMC 12A·Medium

Isosceles triangles

T

and

T'

are not congruent but have the same area and the same perimeter. The sides of

T

have lengths

5

,

5

, and

8

, while those of

T'

have lengths

a

,

a

, and

b

. Which of the following numbers is closest to

b

?

AMC 12·2015 · AMC 12A·Hard

A circle of radius

r

passes through both foci of, and exactly four points on, the ellipse with equation

x^2+16y^2=16

. The set of all possible values of

r

is an interval

[a,b)

. What is

a+b

?

AMC 12·2015 · AMC 12A·Hard

A collection of circles in the upper half-plane, all tangent to the

x

-axis, is constructed in layers as follows. Layer

L_0

consists of two circles of radii

70^2

and

73^2

that are externally tangent. For

k\ge1

, the circles in

\bigcup_{j=0}^{k-1}L_j

are ordered according to their points of tangency with the

x

-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer

L_k

consists of the

2^{k-1}

circles constructed in this way. Let

S=\bigcup_{j=0}^{6}L_j

, and for every circle

C

denote by

r(C)

its radius. What is \sum_{C\in S} \frac{1}{\sqrt{r(C)}}?

AMC 12·2015 · AMC 12A·Easy

Two right circular cylinders have the same volume. The radius of the second cylinder is

10\%

more than the radius of the first. What is the relationship between the heights of the two cylinders?

AMC 12·2015 · AMC 12A·Easy

The ratio of the length to the width of a rectangle is

4

:

3

. If the rectangle has diagonal of length

d

, then the area may be expressed as

kd^2

for some constant

k

. What is

k

?

AMC 12·2015 · AMC 12B·Easy

How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?

AMC 12·2015 · AMC 12B·Medium

The line

12x+5y=60

forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

AMC 12·2015 · AMC 12B·Medium

Quadrilateral

ABCD

is inscribed in a circle with

\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,

and

BC=6

. What is

AC

?

AMC 12·2015 · AMC 12B·Medium

A circle of radius 2 is centered at

A

. An equilateral triangle with side 4 has a vertex at

A

. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?

AMC 12·2015 · AMC 12B·Medium

A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

AMC 12·2015 · AMC 12B·Medium

In

\triangle ABC

,

\angle C = 90^\circ

and

AB = 12

. Squares

ABXY

and

CBWZ

are constructed outside of the triangle. The points

X

,

Y

,

Z

, and

W

lie on a circle. What is the perimeter of the triangle?

AMC 12·2015 · AMC 12B·Hard

A rectangular box measures

a \times b \times c

, where

a

,

b

, and

c

are integers and

1\leq a \leq b \leq c

. The volume and the surface area of the box are numerically equal. How many ordered triples

(a,b,c)

are possible?

AMC 12·2015 · AMC 12B·Hard

Four circles, no two of which are congruent, have centers at

A

,

B

,

C

, and

D

, and points

P

and

Q

lie on all four circles. The radius of circle

A

is

\tfrac{5}{8}

times the radius of circle

B

, and the radius of circle

C

is

\tfrac{5}{8}

times the radius of circle

D

. Furthermore,

AB = CD = 39

and

PQ = 48

. Let

R

be the midpoint of

\overline{PQ}

. What is

\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}

?

AMC 12·2015 · AMC 12B·Easy

A regular 15-gon has

L

lines of symmetry, and the smallest positive angle for which it has rotational symmetry is

R

degrees. What is

L+R

?

AMC 12·2016 · AMC 12A·Medium

In

\triangle ABC

,

AB = 6

,

BC = 7

, and

CA = 8

. Point

D

lies on

\overline{BC}

, and

\overline{AD}

bisects

\angle BAC

. Point

E

lies on

\overline{AC}

, and

\overline{BE}

bisects

\angle ABC

. The bisectors intersect at

F

. What is the ratio

AF

:

FD

?

AMC 12·2016 · AMC 12A·Medium

Circles with centers

P, Q

and

R

, having radii

1, 2

and

3

, respectively, lie on the same side of line

l

and are tangent to

l

at

P', Q'

and

R'

, respectively, with

Q'

between

P'

and

R'

. The circle with center

Q

is externally tangent to each of the other two circles. What is the area of triangle

PQR

?

AMC 12·2016 · AMC 12A·Medium

Let

ABCD

be a square. Let

E, F, G

and

H

be the centers, respectively, of equilateral triangles with bases

\overline{AB}, \overline{BC}, \overline{CD},

and

\overline{DA},

each exterior to the square. What is the ratio of the area of square

EFGH

to the area of square

ABCD

?

AMC 12·2016 · AMC 12A·Hard

A quadrilateral is inscribed in a circle of radius

200\sqrt{2}.

Three of the sides of this quadrilateral have length

200.

What is the length of its fourth side?

AMC 12·2016 · AMC 12A·Easy

What is the area of the shaded region of the given

8\times 5

rectangle?

AMC 12·2016 · AMC 12A·Easy

The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is

\tfrac{a-\sqrt{2}}{b}

, where

a

and

b

are positive integers. What is

a+b

?

AMC 12·2016 · AMC 12B·Easy

A quadrilateral has vertices

P(a,b)

,

Q(b,a)

,

R(-a, -b)

, and

S(-b, -a)

, where

a

and

b

are integers with

a>b>0

. The area of

PQRS

is

16

. What is

a+b

?

AMC 12·2016 · AMC 12B·Medium

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line

y=\pi x

, the line

y=-0.1

and the line

x=5.1?

AMC 12·2016 · AMC 12B·Medium

All the numbers

2, 3, 4, 5, 6, 7

are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

AMC 12·2016 · AMC 12B·Medium

In

\triangle ABC

shown in the figure,

AB=7

,

BC=8

,

CA=9

, and

\overline{AH}

is an altitude. Points

D

and

E

lie on sides

\overline{AC}

and

\overline{AB}

, respectively, so that

\overline{BD}

and

\overline{CE}

are angle bisectors, intersecting

\overline{AH}

at

Q

and

P

, respectively. What is

PQ

?

AMC 12·2016 · AMC 12B·Medium

What is the area of the region enclosed by the graph of the equation

x^2+y^2=|x|+|y|?

AMC 12·2016 · AMC 12B·Hard

Let

ABCD

be a unit square. Let

Q_1

be the midpoint of

\overline{CD}

. For

i=1,2,\dots,

let

P_i

be the intersection of

\overline{AQ_i}

and

\overline{BD}

, and let

Q_{i+1}

be the foot of the perpendicular from

P_i

to

\overline{CD}

. What is

\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?

AMC 12·2016 · AMC 12B·Hard

What is the volume of the region in three-dimensional space defined by the inequalities

|x|+|y|+|z|\le1

and

|x|+|y|+|z-1|\le1

?

AMC 12·2016 · AMC 12B·Easy

All three vertices of

\bigtriangleup ABC

lie on the parabola defined by

y=x^2

, with

A

at the origin and

\overline{BC}

parallel to the

x

-axis. The area of the triangle is

64

. What is the length of

BC

?

AMC 12·2016 · AMC 12B·Easy

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length

3

inches weighs

12

ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of

5

inches. Which of the following is closest to the weight, in ounces, of the second piece?

AMC 12·2017 · AMC 12A·Medium

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of

2017

. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

AMC 12·2017 · AMC 12A·Medium

In the figure below, semicircles with centers at

A

and

B

and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter

JK

. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at

P

is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at

P

?

AMC 12·2017 · AMC 12A·Medium

A square with side length

x

is inscribed in a right triangle with sides of length

3

,

4

, and

5

so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length

y

is inscribed in another right triangle with sides of length

3

,

4

, and

5

so that one side of the square lies on the hypotenuse of the triangle. What is

\frac{x}{y}

?

AMC 12·2017 · AMC 12A·Hard

Quadrilateral

ABCD

is inscribed in circle

O

and has side lengths

AB=3, BC=2, CD=6

, and

DA=8

. Let

X

and

Y

be points on

\overline{BD}

such that

\frac{DX}{BD} = \frac{1}{4}

and

\frac{BY}{BD} = \frac{11}{36}

. Let

E

be the intersection of line

AX

and the line through

Y

parallel to

\overline{AD}

. Let

F

be the intersection of line

CX

and the line through

E

parallel to

\overline{AC}

. Let

G

be the point on circle

O

other than

C

that lies on line

CX

. What is

XF\cdot XG

?

AMC 12·2017 · AMC 12A·Easy

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?

AMC 12·2017 · AMC 12A·Easy

Joy has

30

thin rods, one each of every integer length from

1 \text{ cm}

through

30 \text{ cm}

. She places the rods with lengths

3 \text{ cm}

,

7 \text{ cm}

, and

15 \text{cm}

on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

AMC 12·2017 · AMC 12A·Easy

The region consisting of all points in three-dimensional space within

3

units of line segment

\overline{AB}

has volume

216 \pi

. What is the length

AB

?

AMC 12·2017 · AMC 12B·Medium

What is the sum of the roots of

z^{12}=64

that have a positive real part?

AMC 12·2017 · AMC 12B·Medium

An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?

AMC 12·2017 · AMC 12B·Medium

Let

ABC

be an equilateral triangle. Extend side

\overline{AB}

beyond

B

to a point

B'

so that

BB'=3AB

. Similarly, extend side

\overline{BC}

beyond

C

to a point

C'

so that

CC'=3BC

, and extend side

\overline{CA}

beyond

A

to a point

A'

so that

AA'=3CA

. What is the ratio of the area of

\triangle A'B'C'

to the area of

\triangle ABC

?

AMC 12·2017 · AMC 12B·Medium

The diameter

AB

of a circle of radius

2

is extended to a point

D

outside the circle so that

BD=3

. Point

E

is chosen so that

ED=5

and line

ED

is perpendicular to line

AD

. Segment

AE

intersects the circle at a point

C

between

A

and

E

. What is the area of

\triangle ABC

?

AMC 12·2017 · AMC 12B·Hard

Quadrilateral

ABCD

has right angles at

B

and

C

,

\triangle ABC \sim \triangle BCD

, and

AB > BC

. There is a point

E

in the interior of

ABCD

such that

\triangle ABC \sim \triangle CEB

and the area of

\triangle AED

is

17

times the area of

\triangle CEB

. What is

\tfrac{AB}{BC}

?

AMC 12·2017 · AMC 12B·Easy

The circle having

(0,0)

and

(8,6)

as the endpoints of a diameter intersects the

x

-axis at a second point. What is the

x

-coordinate of this point?

AMC 12·2017 · AMC 12B·Easy

The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?

AMC 12·2017 · AMC 12B·Easy

A circle has center

(-10, -4)

and has radius

13

. Another circle has center

(3, 9)

and radius

\sqrt{65}

. The line passing through the two points of intersection of the two circles has equation

x+y=c

. What is

c

?

AMC 12·2018 · AMC 12A·Medium

A paper triangle with sides of lengths

3,4,

and

5

inches, as shown, is folded so that point

A

falls on point

B

. What is the length in inches of the crease?

AMC 12·2018 · AMC 12A·Medium

A scanning code consists of a

7 \times 7

grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of

49

squares. A scanning code is called

\textit{symmetric}

if its look does not change when the entire square is rotated by a multiple of

90 ^{\circ}

counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?

AMC 12·2018 · AMC 12A·Medium

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths

3

and

4

units. In the corner where those sides meet at a right angle, he leaves a small unplanted square

S

so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from

S

to the hypotenuse is

2

units. What fraction of the field is planted?

AMC 12·2018 · AMC 12A·Medium

Triangle

ABC

with

AB=50

and

AC=10

has area

120

. Let

D

be the midpoint of

\overline{AB}

, and let

E

be the midpoint of

\overline{AC}

. The angle bisector of

\angle BAC

intersects

\overline{DE}

and

\overline{BC}

at

F

and

G

, respectively. What is the area of quadrilateral

FDBG

?

AMC 12·2018 · AMC 12A·Medium

Triangle

ABC

is an isosceles right triangle with

AB=AC=3

. Let

M

be the midpoint of hypotenuse

\overline{BC}

. Points

I

and

E

lie on sides

\overline{AC}

and

\overline{AB}

, respectively, so that

AI>AE

and

AIME

is a cyclic quadrilateral. Given that triangle

EMI

has area

2

, the length

CI

can be written as

\frac{a-\sqrt{b}}{c}

, where

a

,

b

, and

c

are positive integers and

b

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

a+b+c

?

AMC 12·2018 · AMC 12A·Hard

In

\triangle PAT, \angle P=36^{\circ}, \angle A=56^{\circ},

and

PA=10.

Points

U

and

G

lie on sides

\overline{TP}

and

\overline{TA},

respectively, so that

PU=AG=1.

Let

M

and

N

be the midpoints of segments

\overline{PA}

and

\overline{UG},

respectively. What is the degree measure of the acute angle formed by lines

MN

and

PA?

AMC 12·2018 · AMC 12A·Easy

All of the triangles in the diagram below are similar to isosceles triangle

ABC

, in which

AB=AC

. Each of the

7

smallest triangles has area

1,

and

\triangle ABC

has area

40

. What is the area of trapezoid

DBCE

?

AMC 12·2018 · AMC 12B·Medium

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point

A

in the figure on the right. The box has base length

w

and height

h

. What is the area of the sheet of wrapping paper?

AMC 12·2018 · AMC 12B·Medium

Side

\overline{AB}

of

\triangle ABC

has length

10

. The bisector of angle

A

meets

\overline{BC}

at

D

, and

CD = 3

. The set of all possible values of

AC

is an open interval

(m,n)

. What is

m+n

?

AMC 12·2018 · AMC 12B·Medium

Square

ABCD

has side length

30

. Point

P

lies inside the square so that

AP = 12

and

BP = 26

. The centroids of

\triangle{ABP}

,

\triangle{BCP}

,

\triangle{CDP}

, and

\triangle{DAP}

are the vertices of a convex quadrilateral. What is the area of that quadrilateral?

AMC 12·2018 · AMC 12B·Medium

The solutions to the equation

(z+6)^8=81

are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled

A,B,

and

C

. What is the least possible area of

\triangle ABC?

AMC 12·2018 · AMC 12B·Medium

Let

ABCDEF

be a regular hexagon with side length

1

. Denote by

X

,

Y

, and

Z

the midpoints of sides

\overline {AB}

,

\overline{CD}

, and

\overline{EF}

, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of

\triangle ACE

and

\triangle XYZ

?

AMC 12·2018 · AMC 12B·Hard

In

\triangle{ABC}

with side lengths

AB = 13

,

AC = 12

, and

BC = 5

, let

O

and

I

denote the circumcenter and incenter, respectively. A circle with center

M

is tangent to the legs

AC

and

BC

and to the circumcircle of

\triangle{ABC}

. What is the area of

\triangle{MOI}

?

AMC 12·2018 · AMC 12B·Hard

Ajay is standing at point

A

near Pontianak, Indonesia,

0^\circ

latitude and

110^\circ \text{ E}

longitude. Billy is standing at point

B

near Big Baldy Mountain, Idaho, USA,

45^\circ \text{ N}

latitude and

115^\circ \text{ W}

longitude. Assume that Earth is a perfect sphere with center

C.

What is the degree measure of

\angle ACB?

AMC 12·2018 · AMC 12B·Hard

Circles

\omega_1

,

\omega_2

, and

\omega_3

each have radius

4

and are placed in the plane so that each circle is externally tangent to the other two. Points

P_1

,

P_2

, and

P_3

lie on

\omega_1

,

\omega_2

, and

\omega_3

respectively such that

P_1P_2=P_2P_3=P_3P_1

and line

P_iP_{i+1}

is tangent to

\omega_i

for each

i=1,2,3

, where

P_4 = P_1

. See the figure below. The area of

\triangle P_1P_2P_3

can be written in the form

\sqrt{a}+\sqrt{b}

for positive integers

a

and

b

. What is

a+b

?

AMC 12·2018 · AMC 12B·Easy

A circle has a chord of length

10

, and the distance from the center of the circle to the chord is

5

. What is the area of the circle?

AMC 12·2018 · AMC 12B·Easy

Line segment

\overline{AB}

is a diameter of a circle with

AB = 24

. Point

C

, not equal to

A

or

B

, lies on the circle. As point

C

moves around the circle, the centroid (center of mass) of

\triangle ABC

traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

AMC 12·2019 · AMC 12A·Easy

The area of a pizza with radius

4

is

N

percent larger than the area of a pizza with radius

3

inches. What is the integer closest to

N

?

AMC 12·2019 · AMC 12A·Easy

The figure below shows

13

circles of radius

1

within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius

1

?

AMC 12·2019 · AMC 12A·Medium

A sphere with center

O

has radius

6

. A triangle with sides of length

15, 15,

and

24

is situated in space so that each of its sides is tangent to the sphere. What is the distance between

O

and the plane determined by the triangle?

AMC 12·2019 · AMC 12A·Medium

In

\triangle ABC

with integer side lengths,

\cos A = \frac{11}{16}

,

\cos B = \frac{7}{8}

, and

\cos C = -\frac{1}{4}

. What is the least possible perimeter for

\triangle ABC

?

AMC 12·2019 · AMC 12A·Hard

Circles

\omega

and

\gamma

, both centered at

O

, have radii

20

and

17

, respectively. Equilateral triangle

ABC

, whose interior lies in the interior of

\omega

but in the exterior of

\gamma

, has vertex

A

on

\omega

, and the line containing side

\overline{BC}

is tangent to

\gamma

. Segments

\overline{AO}

and

\overline{BC}

intersect at

P

, and

\dfrac{BP}{CP} = 3

. Then

AB

can be written in the form

\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}

for positive integers

m

,

n

,

p

,

q

with

\gcd(m,n) = \gcd(p,q) = 1

. What is

m+n+p+q

?

\phantom{}

AMC 12·2019 · AMC 12A·Hard

Let

\triangle A_0B_0C_0

be a triangle whose angle measures are exactly

59.999^\circ

,

60^\circ

, and

60.001^\circ

. For each positive integer

n

, define

A_n

to be the foot of the altitude from

A_{n-1}

to line

B_{n-1}C_{n-1}

. Likewise, define

B_n

to be the foot of the altitude from

B_{n-1}

to line

A_{n-1}C_{n-1}

, and

C_n

to be the foot of the altitude from

C_{n-1}

to line

A_{n-1}B_{n-1}

. What is the least positive integer

n

for which

\triangle A_nB_nC_n

is obtuse?

AMC 12·2019 · AMC 12A·Easy

Two lines with slopes

\dfrac{1}{2}

and

2

intersect at

(2,2)

. What is the area of the triangle enclosed by these two lines and the line

x+y=10

?

AMC 12·2019 · AMC 12A·Easy

The figure below shows line

\ell

with a regular, infinite, recurring pattern of squares and line segments. How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? *some rotation around a point of line

\ell

*some translation in the direction parallel to line

\ell

*the reflection across line

\ell

*some reflection across a line perpendicular to line

\ell

AMC 12·2019 · AMC 12B·Easy

Alicia had two containers. The first was

\tfrac{5}{6}

full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was

\tfrac{3}{4}

full of water. What is the ratio of the volume of the first container to the volume of the second container?

AMC 12·2019 · AMC 12B·Medium

How many unordered pairs of edges of a given cube determine a plane?

AMC 12·2019 · AMC 12B·Medium

Right triangle

ACD

with right angle at

C

is constructed outwards on the hypotenuse

\overline{AC}

of isosceles right triangle

ABC

with leg length

1

, as shown, so that the two triangles have equal perimeters. What is

\sin(2\angle BAD)

?

AMC 12·2019 · AMC 12B·Medium

As shown in the figure, line segment

\overline{AD}

is trisected by points

B

and

C

so that

AB=BC=CD=2.

Three semicircles of radius

1, \overset{\frown}{AEB},\overset{\frown}{BFC},

and

\overset{\frown}{CGD},

have their diameters on

\overline{AD},

and are tangent to line

EG

at

E,F,

and

G,

respectively. A circle of radius

2

has its center on

F.

The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \frac{a}{b}\cdot\pi-\sqrt{c}+d, where

a,b,c,

and

d

are positive integers and

a

and

b

are relatively prime. What is

a+b+c+d

?

AMC 12·2019 · AMC 12B·Medium

How many nonzero complex numbers

z

have the property that

0, z,

and

z^3,

when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

AMC 12·2019 · AMC 12B·Medium

Square pyramid

ABCDE

has base

ABCD

, which measures

3

cm on a side, and altitude

AE

perpendicular to the base, which measures

6

cm. Point

P

lies on

BE

, one third of the way from

B

to

E

; point

Q

lies on

DE

, one third of the way from

D

to

E

; and point

R

lies on

CE

, two thirds of the way from

C

to

E

. What is the area, in square centimeters, of

\triangle{PQR}

?

AMC 12·2019 · AMC 12B·Medium

Points

A(6,13)

and

B(12,11)

lie on circle

\omega

in the plane. Suppose that the tangent lines to

\omega

at

A

and

B

intersect at a point on the

x

-axis. What is the area of

\omega

?

AMC 12·2019 · AMC 12B·Hard

Let

\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.

Let

S

denote all points in the complex plane of the form

a+b\omega+c\omega^2,

where

0\leq a \leq 1,0\leq b\leq 1,

and

0\leq c\leq 1.

What is the area of

S

?

AMC 12·2019 · AMC 12B·Hard

Let

ABCD

be a convex quadrilateral with

BC=2

and

CD=6.

Suppose that the centroids of

\triangle ABC,\triangle BCD,

and

\triangle ACD

form the vertices of an equilateral triangle. What is the maximum possible value of the area of

ABCD

?

AMC 12·2019 · AMC 12B·Easy

Which of the following rigid transformations (isometries) maps the line segment

\overline{AB}

onto the line segment

\overline{A'B'}

so that the image of

A(-2, 1)

is

A'(2, -1)

and the image of

B(-1, 4)

is

B'(1, -4)

?

AMC 12·2019 · AMC 12B·Easy

In a given plane, points

A

and

B

are

10

units apart. How many points

C

are there in the plane such that the perimeter of

\triangle ABC

is

50

units and the area of

\triangle ABC

is

100

square units?

AMC 12·2019 · AMC 12B·Easy

For how many integral values of

x

can a triangle of positive area be formed having side lengths

\log_{2} x, \log_{4} x, 3

?

AMC 12·2020 · AMC 12A·Medium

Line

l

in the coordinate plane has equation

3x-5y+40=0

. This line is rotated

45^{\circ}

counterclockwise about the point

(20,20)

to obtain line

k

. What is the

x

-coordinate of the

x

-intercept of line

k?

AMC 12·2020 · AMC 12A·Medium

Regular octagon

ABCDEFGH

has area

n

. Let

m

be the area of quadrilateral

ACEG

. What is

\tfrac{m}{n}?

AMC 12·2020 · AMC 12A·Medium

The vertices of a quadrilateral lie on the graph of

y = \ln x

, and the

x

-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is

\ln \frac{91}{90}

. What is the

x

-coordinate of the leftmost vertex?

AMC 12·2020 · AMC 12A·Medium

Quadrilateral

ABCD

satisfies

\angle ABC = \angle ACD = 90^{\circ}, AC = 20

, and

CD = 30

. Diagonals

\overline{AC}

and

\overline{BD}

intersect at point

E

, and

AE = 5

. What is the area of quadrilateral

ABCD

?

AMC 12·2020 · AMC 12A·Medium

Let

T

be the triangle in the coordinate plane with vertices

\left(0,0\right)

,

\left(4,0\right)

, and

\left(0,3\right)

. Consider the following five isometries (rigid transformations) of the plane: rotations of

90^{\circ}

,

180^{\circ}

, and

270^{\circ}

counterclockwise around the origin, reflection across the

x

-axis, and reflection across the

y

-axis. How many of the

125

sequences of three of these transformations (not necessarily distinct) will return

T

to its original position?

AMC 12·2020 · AMC 12A·Hard

Suppose that

\triangle{ABC}

is an equilateral triangle of side length

s

, with the property that there is a unique point

P

inside the triangle such that

AP=1

,

BP=\sqrt{3}

, and

CP=2

. What is

s

?

AMC 12·2020 · AMC 12A·Easy

Seven cubes, whose volumes are

1

,

8

,

27

,

64

,

125

,

216

, and

343

cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

AMC 12·2020 · AMC 12B·Easy

In unit square

ABCD,

the inscribed circle

\omega

intersects

\overline{CD}

at

M,

and

\overline{AM}

intersects

\omega

at a point

P

different from

M.

What is

AP?

AMC 12·2020 · AMC 12B·Medium

Let

\overline{AB}

be a diameter in a circle of radius

5\sqrt2.

Let

\overline{CD}

be a chord in the circle that intersects

\overline{AB}

at a point

E

such that

BE=2\sqrt5

and

\angle AEC = 45^{\circ}.

What is

CE^2+DE^2?

AMC 12·2020 · AMC 12B·Medium

In square

ABCD

, points

E

and

H

lie on

\overline{AB}

and

\overline{DA}

, respectively, so that

AE=AH.

Points

F

and

G

lie on

\overline{BC}

and

\overline{CD}

, respectively, and points

I

and

J

lie on

\overline{EH}

so that

\overline{FI} \perp \overline{EH}

and

\overline{GJ} \perp \overline{EH}

. See the figure below. Triangle

AEH

, quadrilateral

BFIE

, quadrilateral

DHJG

, and pentagon

FCGJI

each has area

1.

What is

FI^2

?

AMC 12·2020 · AMC 12B·Medium

Square

ABCD

in the coordinate plane has vertices at the points

A(1,1), B(-1,1), C(-1,-1),

and

D(1,-1).

Consider the following four transformations:

\bullet L,

a rotation of

90^{\circ}

counterclockwise around the origin;

\bullet R,

a rotation of

90^{\circ}

clockwise around the origin;

\bullet H,

a reflection across the

x

-axis; and

\bullet V,

a reflection across the

y

-axis. Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying

R

and then

V

would send the vertex

A

at

(1,1)

to

(-1,-1)

and would send the vertex

B

at

(-1,1)

to itself. How many sequences of

20

transformations chosen from

\{L, R, H, V\}

will send all of the labeled vertices back to their original positions? (For example,

R, R, V, H

is one sequence of

4

transformations that will send the vertices back to their original positions.)

AMC 12·2020 · AMC 12B·Hard

How many integers

n \geq 2

are there such that whenever

z_1, z_2, ..., z_n

are complex numbers such that

|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,

then the numbers

z_1, z_2, ..., z_n

are equally spaced on the unit circle in the complex plane?

AMC 12·2020 · AMC 12B·Easy

The acute angles of a right triangle are

a^{\circ}

and

b^{\circ}

, where

a>b

and both

a

and

b

are prime numbers. What is the least possible value of

b

?

AMC 12·2020 · AMC 12B·Easy

A three-quarter sector of a circle of radius

4

inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

AMC 12·2021 · AMC 12A·Hard

Semicircle

\Gamma

has diameter

\overline{AB}

of length

14

. Circle

\Omega

lies tangent to

\overline{AB}

at a point

P

and intersects

\Gamma

at points

Q

and

R

. If

QR=3\sqrt3

and

\angle QPR=60^\circ

, then the area of

\triangle PQR

equals

\tfrac{a\sqrt{b}}{c}

, where

a

and

c

are relatively prime positive integers, and

b

is a positive integer not divisible by the square of any prime. What is

a+b+c

?

AMC 12·2021 · AMC 12B·Medium

Triangle

ABC

has

AB=13,BC=14

and

AC=15

. Let

P

be the point on

\overline{AC}

such that

PC=10

. There are exactly two points

D

and

E

on line

BP

such that quadrilaterals

ABCD

and

ABCE

are trapezoids. What is the distance

DE?

AMC 12·2021 · AMC 12B·Medium

Let

ABCD

be a rectangle and let

\overline{DM}

be a segment perpendicular to the plane of

ABCD

. Suppose that

\overline{DM}

has integer length, and the lengths of

\overline{MA},\overline{MC},

and

\overline{MB}

are consecutive odd positive integers (in this order). What is the volume of pyramid

MABCD?

AMC 12·2021 · AMC 12B·Medium

Let

ABCD

be an isoceles trapezoid having parallel bases

\overline{AB}

and

\overline{CD}

with

AB>CD.

Line segments from a point inside

ABCD

to the vertices divide the trapezoid into four triangles whose areas are

2, 3, 4,

and

5

starting with the triangle with base

\overline{CD}

and moving clockwise as shown in the diagram below. What is the ratio

\frac{AB}{CD}?

AMC 12·2021 · AMC 12B·Hard

Let

ABCD

be a parallelogram with area

15

. Points

P

and

Q

are the projections of

A

and

C,

respectively, onto the line

BD;

and points

R

and

S

are the projections of

B

and

D,

respectively, onto the line

AC.

See the figure, which also shows the relative locations of these points. Suppose

PQ=6

and

RS=8,

and let

d

denote the length of

\overline{BD},

the longer diagonal of

ABCD.

Then

d^2

can be written in the form

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 12·2021 · AMC 12A Fall·Medium

Consider two concentric circles of radius

17

and

19.

The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?

AMC 12·2021 · AMC 12A Fall·Medium

The angle bisector of the acute angle formed at the origin by the graphs of the lines

y = x

and

y=3x

has equation

y=kx.

What is

k?

AMC 12·2021 · AMC 12A Fall·Medium

In the figure, equilateral hexagon

ABCDEF

has three nonadjacent acute interior angles that each measure

30^\circ

. The enclosed area of the hexagon is

6\sqrt{3}

. What is the perimeter of the hexagon?

AMC 12·2021 · AMC 12A Fall·Hard

Let

ABCD

be an isosceles trapezoid with

\overline{BC}\parallel \overline{AD}

and

AB=CD

. Points

X

and

Y

lie on diagonal

\overline{AC}

with

X

between

A

and

Y

, as shown in the figure. Suppose

\angle AXD = \angle BYC = 90^\circ

,

AX = 3

,

XY = 1

, and

YC = 2

. What is the area of

ABCD?

AMC 12·2021 · AMC 12A Fall·Easy

A right rectangular prism whose surface area and volume are numerically equal has edge lengths

\log_{2}x, \log_{3}x,

and

\log_{4}x.

What is

x?

AMC 12·2021 · AMC 12B Fall·Hard

Right triangle

ABC

has side lengths

BC=6

,

AC=8

, and

AB=10

. A circle centered at

O

is tangent to line

BC

at

B

and passes through

A

. A circle centered at

P

is tangent to line

AC

at

A

and passes through

B

. What is

OP

?

AMC 12·2021 · AMC 12B Fall·Hard

Triangle

ABC

has side lengths

AB = 11, BC=24

, and

CA = 20

. The bisector of

\angle{BAC}

intersects

\overline{BC}

in point

D

, and intersects the circumcircle of

\triangle{ABC}

in point

E \ne A

. The circumcircle of

\triangle{BED}

intersects the line

AB

in points

B

and

F \ne B

. What is

CF

?

AMC 12·2021 · AMC 12B Fall·Easy

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?

AMC 12·2021 · AMC 12B Fall·Easy

Triangle

ABC

is equilateral with side length

6

. Suppose that

O

is the center of the inscribed circle of this triangle. What is the area of the circle passing through

A

,

O

, and

C

?

AMC 12·2022 · AMC 12A·Medium

Let

M

be the midpoint of

\overline{AB}

in regular tetrahedron

ABCD

. What is

\cos(\angle CMD)

?

AMC 12·2022 · AMC 12A·Medium

Let

\mathcal{R}

be the region in the complex plane consisting of all complex numbers

z

that can be written as the sum of complex numbers

z_1

and

z_2

, where

z_1

lies on the segment with endpoints

3

and

4i

, and

z_2

has magnitude at most

1

. What integer is closest to the area of

\mathcal{R}

?

AMC 12·2022 · AMC 12A·Medium

Isosceles trapezoid

ABCD

has parallel sides

\overline{AD}

and

\overline{BC},

with

BC < AD

and

AB = CD.

There is a point

P

in the plane such that

PA=1, PB=2, PC=3,

and

PD=4.

What is

\tfrac{BC}{AD}?

AMC 12·2022 · AMC 12A·Hard

Let

c

be a real number, and let

z_1

and

z_2

be the two complex numbers satisfying the equation

z^2 - cz + 10 = 0

. Points

z_1

,

z_2

,

\frac{1}{z_1}

, and

\frac{1}{z_2}

are the vertices of (convex) quadrilateral

\mathcal{Q}

in the complex plane. When the area of

\mathcal{Q}

obtains its maximum possible value,

c

is closest to which of the following?

AMC 12·2022 · AMC 12A·Hard

A circle with integer radius

r

is centered at

(r, r)

. Distinct line segments of length

c_i

connect points

(0, a_i)

to

(b_i, 0)

for

1 \le i \le 14

and are tangent to the circle, where

a_i

,

b_i

, and

c_i

are all positive integers and

c_1 \le c_2 \le \cdots \le c_{14}

. What is the ratio

\frac{c_{14}}{c_1}

for the least possible value of

r

?

AMC 12·2022 · AMC 12A·Easy

Five rectangles,

A

,

B

,

C

,

D

, and

E

, are arranged in a square as shown below. These rectangles have dimensions

1\times6

,

2\times4

,

5\times6

,

2\times7

, and

2\times3

, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?

AMC 12·2022 · AMC 12B·Easy

Regular hexagon

ABCDEF

has side length

2

. Let

G

be the midpoint of

\overline{AB}

, and let

H

be the midpoint of

\overline{DE}

. What is the perimeter of

GCHF

?

AMC 12·2022 · AMC 12B·Medium

The diagram below shows a rectangle with side lengths

4

and

8

and a square with side length

5

. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

AMC 12·2022 · AMC 12B·Medium

The graph of

y=x^2+2x-15

intersects the

x

-axis at points

A

and

C

and the

y

-axis at point

B

. What is

\tan(\angle ABC)

?

AMC 12·2022 · AMC 12B·Medium

In

\triangle{ABC}

medians

\overline{AD}

and

\overline{BE}

intersect at

G

and

\triangle{AGE}

is equilateral. Then

\cos(C)

can be written as

\frac{m\sqrt p}n

, where

m

and

n

are relatively prime positive integers and

p

is a positive integer not divisible by the square of any prime. What is

m+n+p?

AMC 12·2022 · AMC 12B·Easy

In rhombus

ABCD

, point

P

lies on segment

\overline{AD}

so that

\overline{BP} \perp \overline{AD}

,

AP = 3

, and

PD = 2

. What is the area of

ABCD

? (Note: The figure is not drawn to scale.)

AMC 12·2022 · AMC 12B·Hard

Let

S

be the set of circles in the coordinate plane that are tangent to each of the three circles with equations

x^{2}+y^{2}=4

,

x^{2}+y^{2}=64

, and

(x-5)^{2}+y^{2}=3

. What is the sum of the areas of all circles in

S

?

AMC 12·2022 · AMC 12B·Hard

The figure below depicts a regular

7

-gon inscribed in a unit circle. What is the sum of the

4

th powers of the lengths of all

21

of its edges and diagonals?

AMC 12·2022 · AMC 12B·Hard

Four regular hexagons surround a square with side length 1, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as

m \sqrt{n} + p

, where

m

,

n

, and

p

are integers and

n

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

m+n+p

?

AMC 12·2022 · AMC 12B·Easy

The point

(-1, -2)

is rotated

270^{\circ}

counterclockwise about the point

(3, 1)

. What are the coordinates of its new position?

AMC 12·2023 · AMC 12A·Medium

What is the value of

2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?

AMC 12·2023 · AMC 12A·Medium

Usain is walking for exercise by zigzagging across a

100

-meter by

30

-meter rectangular field, beginning at point

A

and ending on the segment

\overline{BC}

. He wants to increase the distance walked by zigzagging as shown in the figure below

(APQRS)

. What angle

\theta = \angle PAB=\angle QPC=\angle RQB=\cdots

will produce a length that is

120

meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)

AMC 12·2023 · AMC 12A·Medium

Circle

C_1

and

C_2

each have radius

1

, and the distance between their centers is

\frac{1}{2}

. Circle

C_3

is the largest circle internally tangent to both

C_1

and

C_2

. Circle

C_4

is internally tangent to both

C_1

and

C_2

and externally tangent to

C_3

. What is the radius of

C_4

?

AMC 12·2023 · AMC 12A·Easy

A square of area

2

is inscribed in a square of area

3

, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?

AMC 12·2023 · AMC 12B·Easy

In the

xy

-plane, a circle of radius

4

with center on the positive

x

-axis is tangent to the

y

-axis at the origin, and a circle with radius

10

with center on the positive

y

-axis is tangent to the

x

-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

AMC 12·2023 · AMC 12B·Medium

What is the maximum area of an isosceles trapezoid that has legs of length

1

and one base twice as long as the other?

AMC 12·2023 · AMC 12B·Medium

A rectangular box

P

has distinct edge lengths

a

,

b

, and

c

. The sum of the lengths of all

12

edges of

P

is

13

, the sum of the areas of all

6

faces of

P

is

\frac{11}{2}

, and the volume of

P

is

\frac{1}{2}

. What is the length of the longest interior diagonal connecting two vertices of

P

?

AMC 12·2023 · AMC 12B·Medium

Triangle

ABC

has side lengths in arithmetic progression, and the smallest side has length

6.

If the triangle has an angle of

120^\circ,

what is the area of

ABC

?

AMC 12·2023 · AMC 12B·Hard

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is

3\sqrt3

inches, its top diameter is

6

inches, and its bottom diameter is

12

inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?

AMC 12·2023 · AMC 12B·Hard

A regular pentagon with area

\sqrt{5}+1

is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

AMC 12·2023 · AMC 12B·Easy

A

3-4-5

right triangle is inscribed in circle

A

, and a

5-12-13

right triangle is inscribed in circle

B

. What is the ratio of the area of circle

A

to the area of circle

B

?

AMC 12·2023 · AMC 12B·Easy

You are playing a game. A

2 \times 1

rectangle covers two adjacent squares (oriented either horizontally or vertically) of a

3 \times 3

grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

AMC 12·2023 · AMC 12B·Easy

What is the area of the region in the coordinate plane defined by

\left||x|-1\right|+\left||y|-1\right|\leq 1?

AMC 12·2024 · AMC 12A·Easy

Let

\alpha

be the radian measure of the smallest angle in a

3{-}4{-}5

right triangle. Let

\beta

be the radian measure of the smallest angle in a

7{-}24{-}25

right triangle. In terms of

\alpha

, what is

\beta

?

AMC 12·2024 · AMC 12A·Medium

The graph of

y=e^{x+1}+e^{-x}-2

has an axis of symmetry. What is the reflection of the point

(-1,\tfrac{1}{2})

over this axis?

AMC 12·2024 · AMC 12A·Medium

Cyclic quadrilateral

ABCD

has lengths

BC=CD=3

and

DA=5

with

\angle CDA=120^\circ

. What is the length of the shorter diagonal of

ABCD

?

AMC 12·2024 · AMC 12A·Hard

A

\textit{disphenoid}

is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?

AMC 12·2024 · AMC 12A·Hard

A graph is

\textit{symmetric}

about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers

(a,b,c,d)

, where

|a|,|b|,|c|,|d|\le5

and

c

and

d

are not both

0

, is the graph of

y=\frac{ax+b}{cx+d}

symmetric about the line

y=x

?

AMC 12·2024 · AMC 12A·Easy

A square and an isosceles triangle are joined along an edge to form a pentagon

10

inches tall and

22

inches wide, as shown below. What is the perimeter of the pentagon, in inches?

AMC 12·2024 · AMC 12A·Easy

In

\Delta ABC

,

\angle ABC = 90^\circ

and

BA = BC = \sqrt{2}

. Points

P_1, P_2, \dots, P_{2024}

lie on hypotenuse

\overline{AC}

so that

AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C

. What is the length of the vector sum

\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?

AMC 12·2024 · AMC 12B·Medium

Suppose

z

is a complex number with positive imaginary part, with real part greater than

1

, and with

|z| = 2

. In the complex plane, the four points

0

,

z

,

z^{2}

, and

z^{3}

are the vertices of a quadrilateral with area

15

. What is the imaginary part of

z

?

AMC 12·2024 · AMC 12B·Medium

A triangle in the coordinate plane has vertices

A(\log_21,\log_22)

,

B(\log_23,\log_24)

, and

C(\log_27,\log_28)

. What is the area of

\triangle ABC

?

AMC 12·2024 · AMC 12B·Medium

Equilateral

\triangle ABC

with side length

14

is rotated about its center by angle

\theta

, where

0 < \theta < 60^{\circ}

, to form

\triangle DEF

. See the figure. The area of hexagon

ADBECF

is

91\sqrt{3}

. What is

\tan\theta

?

AMC 12·2024 · AMC 12B·Medium

Suppose

A

,

B

, and

C

are points in the plane with

AB=40

and

AC=42

, and let

x

be the length of the line segment from

A

to the midpoint of

\overline{BC}

. Define a function

f

by letting

f(x)

be the area of

\triangle ABC

. Then the domain of

f

is an open interval

(p,q)

, and the maximum value

r

of

f(x)

occurs at

x=s

. What is

p+q+r+s

?

AMC 12·2024 · AMC 12B·Hard

The measures of the smallest angles of three different right triangles sum to

90^\circ

. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are

3-4-5

and

5-12-13

. What is the perimeter of the third triangle?

AMC 12·2024 · AMC 12B·Hard

Let

\triangle ABC

be a triangle with integer side lengths and the property that

\angle{B} = 2\angle{A}

. What is the least possible perimeter of such a triangle?

AMC 12·2024 · AMC 12B·Hard

A right pyramid has regular octagon

ABCDEFGH

with side length

1

as its base and apex

V.

Segments

\overline{AV}

and

\overline{DV}

are perpendicular. What is the square of the height of the pyramid?

AMC 12·2024 · AMC 12B·Hard

What is the number of ordered triples

(a,b,c)

of positive integers, with

a\le b\le c\le 9

, such that there exists a (non-degenerate) triangle

\triangle ABC

with an integer inradius for which

a

,

b

, and

c

are the lengths of the altitudes from

A

to

\overline{BC}

,

B

to

\overline{AC}

, and

C

to

\overline{AB}

, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)

AMC 12·2024 · AMC 12B·Easy

In the figure below

WXYZ

is a rectangle with

WX=4

and

WZ=8

. Point

M

lies

\overline{XY}

, point

A

lies on

\overline{YZ}

, and

\angle WMA

is a right angle. The areas of

\triangle WXM

and

\triangle WAZ

are equal. What is the area of

\triangle WMA

?

AMC 12·2025 · AMC 12A·Easy

In the figure shown below, major arc

\widehat{AD}

and minor arc

\widehat{BC}

have the same center,

O

. Also,

A

lies between

O

and

B

, and

D

lies between

O

and

C

. Major arc

\widehat{AD}

, minor arc

\widehat{BC}

, and each of the two segments

\overline{AB}

and

\overline{CD}

have length

2\pi

. What is the distance from

O

to

A

?

AMC 12·2025 · AMC 12A·Medium

The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are

A(2,31), B(8,27),

and

C(18,27)

?

AMC 12·2025 · AMC 12A·Medium

Points

F

,

G

, and

H

are collinear with

G

between

F

and

H

. The ellipse with foci at

G

and

H

is internally tangent to the ellipse with foci at

F

and

G

, as shown below. The two ellipses have the same eccentricity

e

, and the ratio of their areas is

2025

. (Recall that the eccentricity of an ellipse is

e = \tfrac{c}{a}

, where

c

is the distance from the center to a focus, and

2a

is the length of the major axis.) What is

e

?

AMC 12·2025 · AMC 12A·Medium

Triangle

\triangle ABC

has side lengths

AB = 80

,

BC = 45

, and

AC = 75

. The bisector

\angle B

and the altitude to side

\overline{AB}

intersect at point

P

. What is

BP

?

AMC 12·2025 · AMC 12A·Medium

The polynomial

(z + i)(z + 2i)(z + 3i) + 10

has three roots in the complex plane, where

i = \sqrt{-1}

. What is the area of the triangle formed by these three roots?

AMC 12·2025 · AMC 12A·Medium

The base of the pentahedron shown below is a

13 \times 8

rectangle, and its lateral faces are two isosceles triangles with base of length

8

and congruent sides of length

13

, and two isosceles trapezoids with bases of length

7

and

13

and nonparallel sides of length

13

. What is the volume of the pentahedron?

AMC 12·2025 · AMC 12A·Hard

A circle of radius

r

is surrounded by

12

circles of radius

1,

externally tangent to the central circle and sequentially tangent to each other, as shown. Then

r

can be written as

\sqrt a + \sqrt b + c,

where

a, b, c

are integers. What is

a+b+c?

AMC 12·2025 · AMC 12A·Easy

In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is

k

, where

0 < k < 1.

The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale). The area of the shaded portion of the figure is

64\%

of the area of the original square. What is

k

?

AMC 12·2025 · AMC 12A·Easy

Pentagon

ABCDE

is inscribed in a circle, and

\angle BEC = \angle CED = 30^\circ

. Let line

AC

and line

BD

intersect at point

F

, and suppose that

AB = 9

and

AD = 24

. What is

BF

?

AMC 12·2025 · AMC 12B·Easy

The altitude to the hypotenuse of a

30{-}60{-}90^\circ

is divided into two segments of lengths

x<y

by the median to the shortest side of the triangle. What is the ratio

\tfrac{x}{x+y}

?

AMC 12·2025 · AMC 12B·Medium

The windshield wiper on the driver's side of a large bus is depicted below. Arm

\overline{AB}

pivots back and forth around point

A

, sweeping out an arc of

60^{\circ}

, symmetric about the vertical line through

A

. The wiper blade

\overline{CD}

is attached to

B

at its midpoint and stays vertical as the arm moves. The arm is

3

feet long, and the wiper blade is

3.5

feet tall. What is the area of the windshield cleaned by the wiper, in square feet, to the nearest hundredth? (Assume that the windshield is a flat vertical surface.)

AMC 12·2025 · AMC 12B·Medium

A container has a

1\times 1

square bottom, a

3 \times {}3

open square top, and four congruent trapezoidal sides, as shown. Starting when the container is empty, a hose that runs water at a constant rate takes

35

minutes to fill the container up to the midline of the trapezoids. How many more minutes will it take to fill the remainder of the container?

AMC 12·2025 · AMC 12B·Medium

A rectangular grid of squares has

141

rows and

91

columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from

1

through

141 \times 91 = 12{,}831

into the squares. Horace fills the grid horizontally: he puts

1

through

91

in order from left to right into row

1

, puts

92

through

182

into row

2

in order from left to right, and continues similarly through row

141

. Vera fills the grid vertically: she puts

1

through

141

in order from top to bottom into column

1

, then

142

through

282

into column

2

in order from top to bottom, and continues similarly through column

91

. How many squares get two copies of the same number?

AMC 12·2025 · AMC 12B·Hard

Two non-congruent triangles have the same area. Each triangle has sides of length

8

and

9

, and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?

AMC 12·2025 · AMC 12B·Hard

What is the greatest possible area of the triangle in the complex plane with vertices

2z

,

(1+i)z

, and

(1-i)z

, where

z

is a complex number satisfying

|4z - 2| = 1

?

AMC 12·2025 · AMC 12B·Hard

Three concentric circles have radii

1,2,3

. An equilateral triangle with side length

s

has one vertex on each circle. What is

s^2?

Geometry