MathDB

1

Math Prize 2017 Problem 20

\sum_{k = 1}^{11} \frac{\sin(2^{k + 4} \pi / 89)}{\sin(2^k \pi / 89)} \, .

19

1

Math Prize 2017 Problem 19

^\circ

.

18

1

Math Prize 2017 Problem 18

Let

x

,

y

, and

z

be nonnegative integers that are less than or equal to 100. Suppose that

x + y + z

,

xy + z

,

x + yz

, and

xyz

are (in some order) four consecutive terms of an arithmetic sequence. Compute the number of such ordered triples

(x, y, z)

.

17

1

Math Prize 2017 Problem 17

Circle

\omega_1

with radius 3 is inscribed in a strip

S

having border lines

a

and

b

. Circle

\omega_2

within

S

with radius 2 is tangent externally to circle

\omega_1

and is also tangent to line

a

. Circle

\omega_3

within

S

is tangent externally to both circles

\omega_1

and

\omega_2

, and is also tangent to line

b

. Compute the radius of circle

\omega_3

.

16

1

Math Prize 2017 Problem 16

Samantha is about to celebrate her sweet 16th birthday. To celebrate, she chooses a five-digit positive integer of the form SWEET, in which the two E's represent the same digit but otherwise the digits are distinct. (The leading digit S can't be 0.) How many such integers are divisible by 16?

15

1

Math Prize 2017 Problem 15

A restricted rook (RR) is a fictional chess piece that can move horizontally or vertically (like a rook), except that each move is restricted to a neighboring square (cell). If RR can only (with at most one exception) move up and to the right, how many possible distinct paths are there to move RR from the bottom left square to the top right square of a standard 8-by-8 chess board? Note that RR may visit some squares more than once. A path is the sequence of squares visited by RR on its way.

14

1

Math Prize 2017 Problem 14

A permutation of a finite set

S

is a one-to-one function from

S

to

S

. Given a permutation

f

of the set

\{ 1, 2, \ldots, 100 \}

, define the displacement of

f

to be the sum

\sum_{i = 1}^{100} \left\lvert f(i) - i \right\rvert .

How many permutations of

\{ 1, 2, \ldots, 100 \}

have displacement 4?

13

1

Math Prize 2017 Problem 13

A polynomial whose roots are all equal to each other is called a unicorn. Compute the number of distinct ordered triples

(M, P, G)

, where

M

,

P

,

G

are complex numbers, such that the polynomials

z^3 + M z^2 + Pz + G

and

z^3 + G z^2 + Pz + M

are both unicorns.

12

1

Math Prize 2017 Problem 12

Let

S

be the set of all real values of

x

with

0 < x < \pi/2

such that

\sin x

,

\cos x

, and

\tan x

form the side lengths (in some order) of a right triangle. Compute the sum of

\tan^2 x

over all

x

in

S

.

11

1

Math Prize 2017 Problem 11

Let

S(N)

be the number of 1's in the binary representation of an integer

N

, and let

D(N) = S(N + 1) - S(N)

. Compute the sum of

D(N)

over all

N

such that

1 \le N \le 2017

and

D(N) < 0

.

10

1

Math Prize 2017 Problem 10

Let

C

be a cube. Let

P

,

Q

, and

R

be random vertices of

C

, chosen uniformly and independently from the set of vertices of

C

. (Note that

P

,

Q

, and

R

might be equal.) Compute the probability that some face of

C

contains

P

,

Q

, and

R

.

9

1

Math Prize 2017 Problem 9

Say that a positive integer

n

is smooth if

\frac{1}{n}

has a terminating decimal expansion. (Note that 1 is smooth.) Compute the value of the infinite series

\sum_n \frac{1}{n^3} \, ,

where

n

ranges over all smooth positive integers.

8

1

Math Prize 2017 Problem 8

Let

c

be a complex number. Suppose there exist distinct complex numbers

r

,

s

, and

t

such that for every complex number

z

, we have

(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).

Compute the number of distinct possible values of

c

.

7

1

Math Prize 2017 Problem 7

Let

a_1

,

a_2

, ... be an infinite sequence of integers such that

0 \le a_k \le k

for every positive integer

k

and such that

2017 = \sum_{k = 1}^\infty a_k \cdot k! \, .

What is the value of the infinite series

\sum_{k = 1}^\infty a_k

?

6

1

Math Prize 2017 Problem 6

Let

b

and

c

be integers chosen randomly (uniformly and independently) from the set

\{ -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \} .

(Note that

b

and

c

can be equal.) What is the probability that the two roots of the quadratic

x^2 + bx + c

are consecutive integers?

5

1

Math Prize 2017 Problem 5

The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns: * A digit that is repeated 3 or more times in a row. For example, 0001 and 5555 are not PINs, but 0010 is a PIN. * A pair of digits that is duplicated. For example, 1212 and 6363 are not PINs, but 1221 and 6633 are PINs. How many distinct possible PINs are there?

4

1

Math Prize 2017 Problem 4

If

\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}

, compute the smallest possible value of

\mathrm{GIRLS}

. Here

\mathrm{MATH}

and

\mathrm{WITH}

are 4-digit numbers and

\mathrm{GIRLS}

is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.

3

1

Math Prize 2017 Problem 3

If

A

and

B

are numbers such that the polynomial

x^{2017} + Ax + B

is divisible by

(x + 1)^2

, what is the value of

B

?

2

1

Math Prize 2017 Problem 2

In the figure below,

BDEF

is a square inscribed in

\triangle ABC

. If

\frac{AB}{BC} = \frac{4}{5}

, what is the area of

BDEF

divided by the area of

\triangle ABC

? [asy] unitsize(20); pair A = (0, 3); pair B = (0, 0); pair C = (4, 0); draw(A -- B -- C -- cycle); real w = 12.0 / 7; pair D = (w, 0); pair E = (w, w); pair F = (0, w); draw(D -- E -- F); dot(Label("

A

", A, NW), A); dot(Label("

B

", B, SW), B); dot(Label("

C

", C, SE), C); dot(Label("

D

", D, S), D); dot(Label("

E

", E, NE), E); dot(Label("

F

", F, W), F); [/asy]

1

Math Prize 2017 Problem 1

A bag contains 4 red marbles, 5 yellow marbles, and 6 blue marbles. Three marbles are to be picked out randomly (without replacement). What is the probability that exactly two of them have the same color?

2017 Math Prize for Girls Problems

Subcontests

Math Prize 2017 Problem 20

Math Prize 2017 Problem 19

Math Prize 2017 Problem 18

Math Prize 2017 Problem 17

Math Prize 2017 Problem 16

Math Prize 2017 Problem 15

Math Prize 2017 Problem 14

Math Prize 2017 Problem 13

Math Prize 2017 Problem 12

Math Prize 2017 Problem 11

Math Prize 2017 Problem 10

Math Prize 2017 Problem 9

Math Prize 2017 Problem 8

Math Prize 2017 Problem 7

Math Prize 2017 Problem 6

Math Prize 2017 Problem 5

Math Prize 2017 Problem 4

Math Prize 2017 Problem 3

Math Prize 2017 Problem 2

Math Prize 2017 Problem 1