MathDB

1

Math Prize 2012 Problem 20

x

strictly between

0

and

\frac{\pi}{2}

that satisfy the equation

\tan(15 x) = 15 \tan(x) .

Call these 6 values

r_1

,

r_2

,

r_3

,

r_4

,

r_5

, and

r_6

. What is the value of the sum

\frac{1}{\tan^2 r_1} + \frac{1}{\tan^2 r_2} + \frac{1}{\tan^2 r_3} + \frac{1}{\tan^2 r_4} + \frac{1}{\tan^2 r_5} + \frac{1}{\tan^2 r_6} \, ?

19

1

Math Prize 2012 Problem 19

Define

L(x) = x - \frac{x^2}{2}

for every real number

x

. If

n

is a positive integer, define

a_n

by

a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),

where there are

n

iterations of

L

. For example,

a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).

As

n

approaches infinity, what value does

n a_n

approach?

18

1

Math Prize 2012 Problem 18

Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability

\frac{1}{2}

. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will take?

17

1

Math Prize 2012 Problem 17

How many ordered triples

(a, b, c)

, where

a

,

b

, and

c

are from the set

\{ 1, 2, 3, \dots, 17 \}

, satisfy the equation

a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2 \, ?

16

1

Math Prize 2012 Problem 16

Say that a complex number

z

is three-presentable if there is a complex number

w

of absolute value

3

such that

z = w - \frac{1}{w}

. Let

T

be the set of all three-presentable complex numbers. The set

T

forms a closed curve in the complex plane. What is the area inside

T

?

15

1

Math Prize 2012 Problem 15

Kate has two bags

X

and

Y

. Bag

X

contains

5

red marbles (and nothing else). Bag

Y

contains

4

red marbles and

1

blue marble (and nothing else). Kate chooses one of her bags at random (each with probability

\frac{1}{2}

) and removes a random marble from that bag (each marble in that bag being equally likely). She repeats the previous step until one of the bags becomes empty. At that point, what is the probability that the blue marble is still in bag

Y

?

14

1

Math Prize 2012 Problem 14

Let

k

be the smallest positive integer such that the binomial coefficient

\binom{10^9}{k}

is less than the binomial coefficient

\binom{10^9 + 1}{k - 1}

. Let

a

be the first (from the left) digit of

k

and let

b

be the second (from the left) digit of

k

. What is the value of

10a + b

?

13

1

Math Prize 2012 Problem 13

For how many integers

n

with

1 \le n \le 2012

is the product

\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)

equal to zero?

12

1

Math Prize 2012 Problem 12

What is the sum of all positive integer values of

n

that satisfy the equation

\cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32} \, ?

11

1

Math Prize 2012 Problem 11

Alison has an analog clock whose hands have the following lengths:

a

inches (the hour hand),

b

inches (the minute hand), and

c

inches (the second hand), with

a < b < c

. The numbers

a

,

b

, and

c

are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day:

A

inches (the hour hand),

B

inches (the minute hand), and

C

inches (the second hand). The numbers

A

,

B

, and

C

(in this order) are consecutive terms of a geometric sequence. What is the value of

\frac{B}{A}

?

10

1

Math Prize 2012 Problem 10

Let

\triangle ABC

be a triangle with a right angle

\angle ABC

. Let

D

be the midpoint of

\overline{BC}

, let

E

be the midpoint of

\overline{AC}

, and let

F

be the midpoint of

\overline{AB}

. Let

G

be the midpoint of

\overline{EC}

. One of the angles of

\triangle DFG

is a right angle. What is the least possible value of

\frac{BC}{AG}

?

9

1

Math Prize 2012 Problem 9

Bianca has a rectangle whose length and width are distinct primes less than 100. Let

P

be the perimeter of her rectangle, and let

A

be the area of her rectangle. What is the least possible value of

\frac{P^2}{A}

?

8

1

Math Prize 2012 Problem 8

Suppose that

x

,

y

, and

z

are real numbers such that

x + y + z = 3

and

x^2 + y^2 + z^2 = 6

. What is the largest possible value of

z

?

7

1

Math Prize 2012 Problem 7

Let

f_1

,

f_2

,

f_3

, ..., be a sequence of numbers such that

f_n = f_{n - 1} + f_{n - 2}

for every integer

n \ge 3

. If

f_7 = 83

, what is the sum of the first 10 terms of the sequence?

6

1

Math Prize 2012 Problem 6

For how many ordered pairs of positive integers

(x, y)

is the least common multiple of

x

and

y

equal to

1{,}003{,}003{,}001

?

5

1

Math Prize 2012 Problem 5

The figure below shows a semicircle inscribed in a right triangle. [asy] draw((0, 0) -- (15, 0) -- (0, 8) -- cycle); real r = 120 / 23; real theta = -aTan(8/15); draw(arc((r, r), r, theta + 180, theta + 360)); [/asy] The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?

4

1

Math Prize 2012 Problem 4

Evaluate the expression

\frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{ 11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)} \, .

3

1

Math Prize 2012 Problem 3

What is the least positive integer

n

such that

n!

is a multiple of

2012^{2012}

?

2

1

Math Prize 2012 Problem 2

In the figure below, the centers of the six congruent circles form a regular hexagon with side length 2. [asy] import graph; pair A = 2dir(0); pair B = 2dir(60); pair C = 2dir(120); pair D = 2dir(180); pair E = 2dir(240); pair F = 2dir(300); path hexagon = A -- B -- C -- D -- E -- F -- cycle; fill(hexagon, gray); filldraw(Circle(A, 1), white); filldraw(Circle(B, 1), white); filldraw(Circle(C, 1), white); filldraw(Circle(D, 1), white); filldraw(Circle(E, 1), white); filldraw(Circle(F, 1), white); draw(hexagon); [/asy] Adjacent circles are tangent to each other. What is the area of the shaded region?

1

Math Prize 2012 Problem 1

In the morning, Esther biked from home to school at an average speed of

x

miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of

x

?

2012 Math Prize For Girls Problems

Subcontests

Math Prize 2012 Problem 20

Math Prize 2012 Problem 19

Math Prize 2012 Problem 18

Math Prize 2012 Problem 17

Math Prize 2012 Problem 16

Math Prize 2012 Problem 15

Math Prize 2012 Problem 14

Math Prize 2012 Problem 13

Math Prize 2012 Problem 12

Math Prize 2012 Problem 11

Math Prize 2012 Problem 10

Math Prize 2012 Problem 9

Math Prize 2012 Problem 8

Math Prize 2012 Problem 7

Math Prize 2012 Problem 6

Math Prize 2012 Problem 5

Math Prize 2012 Problem 4

Math Prize 2012 Problem 3

Math Prize 2012 Problem 2

Math Prize 2012 Problem 1