MathDB

1

Math Prize 2016 Problem 20

a_1

,

a_2

,

a_3

,

a_4

, and

a_5

be random integers chosen independently and uniformly from the set

\{ 0, 1, 2, \dots, 23 \}

. (Note that the integers are not necessarily distinct.) Find the probability that

\sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0.

(Here

\operatorname{cis} \theta

means

\cos \theta + i \sin \theta

.)

19

1

Math Prize 2016 Problem 19

In the coordinate plane, consider points

A = (0, 0)

,

B = (11, 0)

, and

C = (18, 0)

. Line

\ell_A

has slope 1 and passes through

A

. Line

\ell_B

is vertical and passes through

B

. Line

\ell_C

has slope

-1

and passes through

C

. The three lines

\ell_A

,

\ell_B

, and

\ell_C

begin rotating clockwise about points

A

,

B

, and

C

, respectively. They rotate at the same angular rate. At any given time, the three lines form a triangle. Determine the largest possible area of such a triangle.

18

1

Math Prize 2016 Problem 18

Let

T = \{ 1, 2, 3, \dots, 14, 15 \}

. Say that a subset

S

of

T

is handy if the sum of all the elements of

S

is a multiple of

5

. For example, the empty set is handy (because its sum is 0) and

T

itself is handy (because its sum is 120). Compute the number of handy subsets of

T

.

17

1

Math Prize 2016 Problem 17

We define the weight

W

of a positive integer as follows:

W(1) = 0

,

W(2) = 1

,

W(p) = 1 + W(p + 1)

for every odd prime

p

,

W(c) = 1 + W(d)

for every composite

c

, where

d

is the greatest proper factor of

c

. Compute the greatest possible weight of a positive integer less than 100.

16

1

Math Prize 2016 Problem 16

Let

A < B < C < D

be positive integers such that every three of them form the side lengths of an obtuse triangle. Compute the least possible value of

D

.

15

1

Math Prize 2016 Problem 15

Let

H

be a convex, equilateral heptagon whose angles measure (in degrees)

168^\circ

,

108^\circ

,

108^\circ

,

168^\circ

,

x^\circ

,

y^\circ

, and

z^\circ

in clockwise order. Compute the number

y

.

14

1

Math Prize 2016 Problem 14

We call a set

X

of real numbers three-averaging if for every two distinct elements

a

and

b

of

X

, there exists an element

c

in

X

(different from both

a

and

b

) such that the number

(a + b + c)/3

also belongs to

X

. For instance, the set

\{ 0, 1008, 2016 \}

is three-averaging. What is the least possible number of elements in a three-averaging set with more than 3 elements?

13

1

Math Prize 2016 Problem 13

Alice, Beth, Carla, Dana, and Eden play a game in which each of them simultaneously closes her eyes and randomly chooses two of the others to point at (one with each hand). A participant loses if she points at someone who points back at her; otherwise, she wins. Find the probability that all five girls win.

12

1

Math Prize 2016 Problem 12

Let

b_1

,

b_2

,

b_3

,

c_1

,

c_2

, and

c_3

be real numbers such that for every real number

x

, we have

x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3).

Compute

b_1 c_1 + b_2 c_2 + b_3 c_3

.

11

1

Math Prize 2016 Problem 11

Compute the number of ordered pairs of complex numbers

(u, v)

such that

uv = 10

and such that the real and imaginary parts of

u

and

v

are integers.

10

1

Math Prize 2016 Problem 10

How many solutions of the equation

\tan x = \tan \tan x

are on the interval

0 \le x \le \tan^{-1} 942

? (Here

\tan^{-1}

means the inverse tangent function, sometimes written

\arctan

.)

9

1

Math Prize 2016 Problem 9

How many distinct lines pass through the point

(0, 2016)

and intersect the parabola

y = x^2

at two lattice points? (A lattice point is a point whose coordinates are integers.)

8

1

Math Prize 2016 Problem 8

A strip is the region between two parallel lines. Let

A

and

B

be two strips in a plane. The intersection of strips

A

and

B

is a parallelogram

P

. Let

A'

be a rotation of

A

in the plane by

60^\circ

. The intersection of strips

A'

and

B

is a parallelogram with the same area as

P

. Let

x^\circ

be the measure (in degrees) of one interior angle of

P

. What is the greatest possible value of the number

x

?

7

1

Math Prize 2016 Problem 7

Let

S

be the set of all real numbers

x

such that

0 \le x \le 2016 \pi

and

\sin x < 3 \sin(x/3)

. The set

S

is the union of a finite number of disjoint intervals. Compute the total length of all these intervals.

6

1

Math Prize 2016 Problem 6

The largest term in the binomial expansion of

(1 + \tfrac{1}{2})^{31}

is of the form

\frac{a}{b}

, where

a

and

b

are relatively prime positive integers. What is the value of

b

? As an example of a binomial expansion, the binomial expansion of an expression of the form

(x + y)^3

is the sum of four terms

x^3 + 3x^2y + 3xy^2 + y^3.

5

1

Math Prize 2016 Problem 5

A permutation of a finite set

S

is a one-to-one function from

S

to

S

. A permutation

P

of the set

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

is called a W-permutation if

P(1) > P(2) < P(3) > P(4) < P(5)

. A permutation of the set

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

is selected at random. Compute the probability that it is a W-permutation.

4

1

Math Prize 2016 Problem 4

Compute the smallest positive integer

n

such that

2016^n

does not divide

2016!

.

3

1

Math Prize 2016 Problem 3

Compute the least possible value of

ABCD - AB \times CD

, where

ABCD

is a 4-digit positive integer, and

AB

and

CD

are 2-digit positive integers. (Here

A

,

B

,

C

, and

D

are digits, possibly equal. Neither

A

nor

C

can be zero.)

2

1

Math Prize 2016 Problem 2

Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M?

1

Math Prize 2016 Problem 1

Let

T

be a triangle with side lengths 3, 4, and 5. If

P

is a point in or on

T

, what is the greatest possible sum of the distances from

P

to each of the three sides of

T

?

2016 Math Prize for Girls Problems

Subcontests

Math Prize 2016 Problem 20

Math Prize 2016 Problem 19

Math Prize 2016 Problem 18

Math Prize 2016 Problem 17

Math Prize 2016 Problem 16

Math Prize 2016 Problem 15

Math Prize 2016 Problem 14

Math Prize 2016 Problem 13

Math Prize 2016 Problem 12

Math Prize 2016 Problem 11

Math Prize 2016 Problem 10

Math Prize 2016 Problem 9

Math Prize 2016 Problem 8

Math Prize 2016 Problem 7

Math Prize 2016 Problem 6

Math Prize 2016 Problem 5

Math Prize 2016 Problem 4

Math Prize 2016 Problem 3

Math Prize 2016 Problem 2

Math Prize 2016 Problem 1