MathDB

1

Math Prize 2023 Problem 9

The ring shown below is made out of 18 congruent regular hexagons. How many ways are there to tile the ring using tiles that consist of two hexagons, each congruent to any one of the 18 in the design, joined edge-to-edge? (The central hexagon, in black, is not to be covered with a tile and the ring cannot be rotated or reflected.)

20

1

Math Prize 2023 Problem 20

Let

f_1(x) = 2\pi \sin (x)

. For

n > 1

, define

f_n(x)

recursively by

f_n(x) = 2\pi \sin(f_{n-1}(x)).

How many intervals

[a, b]

are there such that \bullet \

0 \le a < b \le 2\pi

, \bullet \

f_6(a) = -2\pi

, \bullet \

f_6(b)=2\pi

, \bullet \ and

f_6

is increasing on

[a, b]

?

19

1

Math Prize 2023 Problem 19

Let

\displaystyle{N = \prod_{k=1}^{1000} (4^k - 1)}

. Determine the largest positive integer

n

such that

5^n

divides evenly into

N

.

18

1

Math Prize 2023 Problem 18

A unit square is decorated with snippets of the graph of

y = x^2

as follows: We consider the graph of

y=x^2

restricted to the domain

0 \le x \le 6

. We cut up the first quadrant (the positive quadrant) into unit squares with lattice vertices. We translate each square so that they are stacked, one on top of the other. We merge all of these squares. How many regions is the unit square divided into by all the overlaid snippets of the graph of the parabola?

17

1

Math Prize 2023 Problem 17

Let

C

be a unit cube. Let

D

be a translate of

C

such that one corner of

D

is located at the center of

C

and one corner of

C

is located at the center of

D

. Let

D^\prime

be the image of

D

under a

60^\circ

clockwise rotation about the line that passes through both cube centers when looking from the center of

D

to the center of

C

. What is the volume of the intersection of

C

with

D^\prime

?

16

1

Math Prize 2023 Problem 16

Let

f(x) = x^2 - 3/4

. Let

f^{(n)}(x)

denote the composition of

f

with itself

n

times. For example,

f^{(3)}(x) = f(f(f(x)))

. Let

R

be the set of complex numbers that is the union of the roots of the polynomials

f^{(n)}(x^2 + 3/4)

over positive integers

n

. Let

B

be the smallest rectangle in the complex plane with sides parallel to the real and imaginary axes that contains

R

. What is the square of the area of

B

?

15

1

Math Prize 2023 Problem 15

A square is divided into four non-overlapping isosceles triangles. Let

X

be the degree measure of one of the twelve angles of these four triangles. Compute the sum of all possible different values of

X

. (Consider all possible diagrams.)

14

1

Math Prize 2023 Problem 14

Five points are chosen uniformly and independently at random on the surface of a sphere. Next, 2 of these 5 points are randomly picked, with every pair equally likely. What is the probability that the 2 points are separated by the plane containing the other 3 points?

13

1

Math Prize 2023 Problem 13

Let

f(t) = \frac{(10 + 9i)t - 10 + 9i}{t + i} ,

where

i=\sqrt{-1}

. Let

P = f(0)

,

Q = f(2023)

, and

R = f(1)

. Determine

\sin^2(m\angle PQR)

.

12

1

Math Prize 2023 Problem 12

Let

S

be the set of fractions of the form

\frac{{\text {lcm}}(A, B)}{A+B}

, where

A

and

B

are positive integers and

{\text{lcm}}(A, B)

is the least common multiple of

A

and

B

. What is the smallest number exceeding 3 in

S

?

11

1

Math Prize 2023 Problem 11

A random triangle is produced as follows. A pair of standard dice is rolled independently three times to get three random numbers between 2 and 12, inclusive, by adding the numbers that come up on each pair rolled. Call these three random numbers

a

,

b

, and

t

. The random triangle has two sides of lengths

a

and

b

with the angle between them measuring

15(t - 1)

degrees. What is the probability that the triangle is a right triangle?

10

1

Math Prize 2023 Problem 10

Find all integers

x

between 0 and the prime number 4099 such that

x^3 - 3

is divisible by 4099.

8

1

Math Prize 2023 Problem 8

For a positive integer

n

, let

p(n)

denote the number of distinct prime numbers that divide evenly into

n

. Determine the number of solutions, in positive integers

n

, to the inequality

\log_4 n \le p(n)

.

7

1

Math Prize 2023 Problem 7

An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive

\sqrt{3}

’s in a row of twelve

\sqrt{3}

’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to

12\sqrt{3}

?

6

1

Math Prize 2023 Problem 6

Solve for

x

: \begin{eqnarray*} v - w + x - y + z & = & 79 \\ v + w + x + y + z & = & -1 \\ v + 2w + 4x + 8y + 16z & = & -2 \\ v + 3w + 9x + 27y + 81z & = & -1 \\ v + 5w + 25x + 125y + 625z & = & 79. \end{eqnarray*}

5

1

Math Prize 2023 Problem 5

Acute triangle

ABC

has area 870. The triangle whose vertices are the feet of the altitudes of

\triangle ABC

has area 48. Determine

\sin^2 A + \sin^2 B + \sin^2 C .

4

1

Math Prize 2023 Problem 4

Let

\triangle A_1A_2A_3

be an equilateral triangle with unit side length. For

k = 1

,

2

, and

3

, let

B_k

be the point on the boundary of

\triangle A_1A_2A_3

located

1/3

unit away from

A_k

in the clockwise direction and let

C_k

be the point on the boundary of

\triangle A_1A_2A_3

located

1/3

unit away from

A_k

in the counterclockwise direction. What fraction of the area of

\triangle A_1A_2A_3

is the area of the intersection of

\triangle B_1B_2B_3

and

\triangle C_1C_2C_3

?

3

1

Math Prize 2023 Problem 3

You have 5000 distinct finite sets. Their intersection is empty. However, the intersection of any two is nonempty. What is the smallest possible number of elements contained in their union?

2

1

Math Prize 2023 Problem 2

In the

xy

-coordinate plane, the horizontal line

y = k

intersects the graph of the cubic

2x^3 + 6x^2 - 4x + 5

in three points

P

,

Q

, and

R

. Given that

Q

is the midpoint of

P

and

R

, what is

k

?

1

Math Prize 2023 Problem 1

The frame of a painting has the form of a

105^{\prime\prime}

by

105^{\prime\prime}

square with a

95^{\prime\prime}

by

95^{\prime\prime}

square removed from its center. The frame is built out of congruent isosceles trapezoids with angles measuring

45^\circ

and

135^\circ

. Each trapezoid has one base on the frame's outer edge and one base on the frame's inner edge. Each outer edge of the frame contains an odd number of trapezoid bases that alternate long, short, long, short, etc. What is the maximum possible number of trapezoids in the frame?

2023 Math Prize for Girls Problems

Subcontests

Math Prize 2023 Problem 9

Math Prize 2023 Problem 20

Math Prize 2023 Problem 19

Math Prize 2023 Problem 18

Math Prize 2023 Problem 17

Math Prize 2023 Problem 16

Math Prize 2023 Problem 15

Math Prize 2023 Problem 14

Math Prize 2023 Problem 13

Math Prize 2023 Problem 12

Math Prize 2023 Problem 11

Math Prize 2023 Problem 10

Math Prize 2023 Problem 8

Math Prize 2023 Problem 7

Math Prize 2023 Problem 6

Math Prize 2023 Problem 5

Math Prize 2023 Problem 4

Math Prize 2023 Problem 3

Math Prize 2023 Problem 2

Math Prize 2023 Problem 1