MathDB

1

Math Prize 2013 Problem 20

a_0

,

a_1

,

a_2

,

\dots

be an infinite sequence of real numbers such that

a_0 = \frac{4}{5}

and

a_{n} = 2 a_{n-1}^2 - 1

for every positive integer

n

. Let

c

be the smallest number such that for every positive integer

n

, the product of the first

n

terms satisfies the inequality

a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.

What is the value of

100c

, rounded to the nearest integer?

19

1

Math Prize 2013 Problem 19

If

n

is a positive integer, let

\phi(n)

be the number of positive integers less than or equal to

n

that are relatively prime to

n

. Compute the value of the infinite sum

\sum_{n=1}^\infty \frac{\phi(n) 2^n}{9^n - 2^n} \, .

18

1

Math Prize 2013 Problem 18

Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let

D

be the number of dice on the table. What is the expected value (average value) of

6^D

?

17

1

Math Prize 2013 Problem 17

Let

f

be the function defined by

f(x) = -2 \sin(\pi x)

. How many values of

x

such that

-2 \le x \le 2

satisfy the equation

f(f(f(x))) = f(x)

?

16

1

Math Prize 2013 Problem 16

If

-3 \le x < \frac{3}{2}

and

x \ne 1

, define

C(x) = \frac{x^3}{1 - x}

. The real root of the cubic

2x^3 + 3x - 7

is of the form

p C^{-1}(q)

, where

p

and

q

are rational numbers. What is the ordered pair

(p, q)

?

15

1

Math Prize 2013 Problem 15

Let

\triangle ABC

be a triangle with

AB = 7

,

BC = 8

, and

AC = 9

. Point

D

is on side

\overline{AC}

such that

\angle CBD

has measure

45^\circ

. What is the length of

\overline{BD}

?

14

1

Math Prize 2013 Problem 14

How many positive integers

n

satisfy the inequality

\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ?

Recall that

\lceil a \rceil

is the least integer that is greater than or equal to

a

.

13

1

Math Prize 2013 Problem 13

Each of

n

boys and

n

girls chooses a random number from the set

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

, uniformly and independently. Let

p_n

be the probability that every boy chooses a different number than every girl. As

n

approaches infinity, what value does

\sqrt[n]{p_n}

approach?

12

1

Math Prize 2013 Problem 12

The rectangular parallelepiped (box)

P

has some special properties. If one dimension of

P

were doubled and another dimension were halved, then the surface area of

P

would stay the same. If instead one dimension of

P

were tripled and another dimension were divided by

3

, then the surface area of

P

would still stay the same. If the middle (by length) dimension of

P

is

1

, compute the least possible volume of

P

.

11

1

Math Prize 2013 Problem 11

Alice throws two standard dice, with

A

being the number on her first die and

B

being the number on her second die. She then draws the line

Ax + By = 2013

. Boris also throws two standard dice, with

C

being the number on his first die and

D

being the number on his second die. He then draws the line

Cx + Dy = 2014

. Compute the probability that these two lines are parallel.

10

1

Math Prize 2013 Problem 10

The following figure shows a walk of length 6: [asy] unitsize(20); for (int x = -5; x <= 5; ++x) for (int y = 0; y <= 5; ++y) dot((x, y)); label("

O

", (0, 0), S); draw((0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- (-1, 1) -- (-1, 2) -- (-1, 3)); [/asy] This walk has three interesting properties:
[*] It starts at the origin, labelled

O

. [*] Each step is 1 unit north, east, or west. There are no south steps. [*] The walk never comes back to a point it has been to.
Let's call a walk with these three properties a northern walk. There are 3 northern walks of length 1 and 7 northern walks of length 2. How many northern walks of length 6 are there?

9

1

Math Prize 2013 Problem 9

Let

A

and

B

be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of

\left| A - B \right|

.

8

1

Math Prize 2013 Problem 8

Let

R

be the set of points

(x, y)

such that

x

and

y

are positive,

x + y

is at most 2013, and

\lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.

Compute the area of set

R

. Recall that

\lfloor a \rfloor

is the greatest integer that is less than or equal to

a

, and

\lceil a \rceil

is the least integer that is greater than or equal to

a

.

7

1

Math Prize 2013 Problem 7

In the figure below,

\triangle ABC

is an equilateral triangle. [asy] import graph; unitsize(60); axes("

x

", "

y

", (0, 0), (1.5, 1.5), EndArrow); real w = sqrt(3) - 1; pair A = (1, 1); pair B = (0, w); pair C = (w, 0); draw(A -- B -- C -- cycle); dot(Label("

A(1, 1)

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

B

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

C

", C, S), C); [/asy] Point

A

has coordinates

(1, 1)

, point

B

is on the positive

y

-axis, and point

C

is on the positive

x

-axis. What is the area of

\triangle ABC

?

6

1

Math Prize 2013 Problem 6

Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.

5

1

Math Prize 2013 Problem 5

Say that a 4-digit positive integer is mixed if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed?

4

1

Math Prize 2013 Problem 4

The MathMatters competition consists of 10 players

P_1

,

P_2

,

\dots

,

P_{10}

competing in a ladder-style tournament. Player

P_{10}

plays a game with

P_9

: the loser is ranked 10th, while the winner plays

P_8

. The loser of that game is ranked 9th, while the winner plays

P_7

. They keep repeating this process until someone plays

P_1

: the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible?

3

1

Math Prize 2013 Problem 3

Let

S_1

,

S_2

,

\dots

,

S_{125}

be 125 sets of 5 numbers each, comprising

625

distinct numbers. Let

m_i

be the median of

S_i

. Let

M

be the median of

m_1

,

m_2

,

\dots

,

m_{125}

. What is the greatest possible number of the 625 numbers that are less than

M

?

2

1

Math Prize 2013 Problem 2

When the binomial coefficient

\binom{125}{64}

is written out in base 10, how many zeros are at the rightmost end?

1

Math Prize 2013 Problem 1

The figure below shows two equilateral triangles each with area 1. [asy] unitsize(40); draw(polygon(3)); draw(rotate(60) * polygon(3)); [/asy] The intersection of the two triangles is a regular hexagon. What is the area of the union of the two triangles?

2013 Math Prize For Girls Problems

Subcontests

Math Prize 2013 Problem 20

Math Prize 2013 Problem 19

Math Prize 2013 Problem 18

Math Prize 2013 Problem 17

Math Prize 2013 Problem 16

Math Prize 2013 Problem 15

Math Prize 2013 Problem 14

Math Prize 2013 Problem 13

Math Prize 2013 Problem 12

Math Prize 2013 Problem 11

Math Prize 2013 Problem 10

Math Prize 2013 Problem 9

Math Prize 2013 Problem 8

Math Prize 2013 Problem 7

Math Prize 2013 Problem 6

Math Prize 2013 Problem 5

Math Prize 2013 Problem 4

Math Prize 2013 Problem 3

Math Prize 2013 Problem 2

Math Prize 2013 Problem 1