MathDB

1

Math Prize 2010 Problem 20

\sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,

where

z

ranges over all 7 solutions (real and nonreal) of the equation

z^7 = -1

?

19

1

Math Prize 2010 Problem 19

Let

S

be the set of 81 points

(x, y)

such that

x

and

y

are integers from

-4

through

4

. Let

A

,

B

, and

C

be random points chosen independently from

S

, with each of the 81 points being equally likely. (The points

A

,

B

, and

C

do not have to be different.) Let

K

be the area of the (possibly degenerate) triangle

ABC

. What is the expected value (average value) of

K^2

?

18

1

Math Prize 2010 Problem 18

If

a

and

b

are positive integers such that

\sqrt{8 + \sqrt{32 + \sqrt{768}}} = a \cos \frac{\pi}{b} \, ,

compute the ordered pair

(a, b)

.

17

1

Math Prize 2010 Problem 17

For every

x \ge -\frac{1}{e}\,

, there is a unique number

W(x) \ge -1

such that

W(x) e^{W(x)} = x.

The function

W

is called Lambert's

W

function. Let

y

be the unique positive number such that

\frac{y}{\log_{2} y} = - \frac{3}{5} \, .

The value of

y

is of the form

e^{-W(z \ln 2)}

for some rational number

z

. What is the value of

z

?

16

1

Math Prize 2010 Problem 16

Let

P

be the quadratic function such that

P(0) = 7

,

P(1) = 10

, and

P(2) = 25

. If

a

,

b

, and

c

are integers such that every positive number

x

less than 1 satisfies

\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},

compute the ordered triple

(a, b, c)

.

15

1

Math Prize 2010 Problem 15

Compute the value of the sum \begin{align*} \frac{1}{1 + \tan^3 0^\circ} &+ \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ} + \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\ &+ \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ} + \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ} \, . \end{align*}

14

1

Math Prize 2010 Problem 14

In the figure below, the three small circles are congruent and tangent to each other. The large circle is tangent to the three small circles. [asy] import graph; unitsize(20); real r = sqrt(3) / 2; filldraw(Circle((0, 0), 1 + r), gray); filldraw(Circle(dir(90), r), white); filldraw(Circle(dir(210), r), white); filldraw(Circle(dir(330), r), white); [/asy] The area of the large circle is 1. What is the area of the shaded region?

13

1

Math Prize 2010 Problem 13

For every positive integer

n

, define

S_n

to be the sum

S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n .

As

n

approaches infinity, what value does

S_n

approach?

12

1

Math Prize 2010 Problem 12

Say that an ordered triple

(a, b, c)

is pleasing if
(a)

a

,

b

, and

c

are in the set

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

, and
(b) both

b - a

and

c - b

are greater than

3

, and at least one of them is equal to

4

.
How many pleasing triples are there?

11

1

Math Prize 2010 Problem 11

In the figure below, each side of the rhombus has length 5 centimeters. [asy] import graph; unitsize(2.0cm); real w = sqrt(3); draw((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- cycle); filldraw(Circle((-0.5, 0), 0.8 / sqrt(pi)), gray); label("

60^\circ

", (w - 0.1, 0), W); [/asy] The circle lies entirely within the rhombus. The area of the circle is

n

square centimeters, where

n

is a positive integer. Compute the number of possible values of

n

.

10

1

Math Prize 2010 Problem 10

The triangle

ABC

lies on the coordinate plane. The midpoint of

\overline{AB}

has coordinates

(-16, -63)

, the midpoint of

\overline{AC}

has coordinates

(13, 50)

, and the midpoint of

\overline{BC}

has coordinates

(6, -85)

. What are the coordinates of point

A

?

9

1

Math Prize 2010 Problem 9

Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?

8

1

Math Prize 2010 Problem 8

When Meena turned 16 years old, her parents gave her a cake with

n

candles, where

n

has exactly 16 different positive integer divisors. What is the smallest possible value of

n

?

7

1

Math Prize 2010 Problem 7

The graph of

{(x^2 + y^2 - 1)}^3 = x^2 y^3

is a heart-shaped curve, shown in the figure below. [asy] import graph; unitsize(10); real f(real x) { return sqrt(cbrt(x^4) - 4 x^2 + 4); } real g(real x) { return (cbrt(x^2) + f(x))/2; } real h(real x) { return (cbrt(x^2) - f(x)) / 2; } real xmax = 1.139028; draw(graph(g, -xmax, xmax) -- reverse(graph(h, -xmax, xmax)) -- cycle); xaxis("

x

", -1.5, 1.5, above = true); yaxis("

y

", -1.5, 1.5, above = true); [/asy] For how many ordered pairs of integers

(x, y)

is the point

(x, y)

inside or on this curve?

6

1

Math Prize 2010 Problem 6

The bases of a trapezoid have lengths 10 and 21, and the legs have lengths

\sqrt{34}

and

3 \sqrt{5}

. What is the area of the trapezoid?

5

1

Math Prize 2010 Problem 5

Find the smallest two-digit positive integer that is a divisor of 201020112012.

4

1

Math Prize 2010 Problem 4

Consider the sequence of six real numbers 60, 10, 100, 150, 30, and

x

. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of

x

? (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)

3

1

Math Prize 2010 Problem 3

How many ordered triples of integers

(x, y, z)

are there such that

x^2 + y^2 + z^2 = 34 \, ?

2

1

Math Prize 2010 Problem 2

Jane has two bags

X

and

Y

. Bag

X

contains 4 red marbles and 5 blue marbles (and nothing else). Bag

Y

contains 7 red marbles and 6 blue marbles (and nothing else). Jane will choose one of her bags at random (each bag being equally likely). From her chosen bag, she will then select one of the marbles at random (each marble in that bag being equally likely). What is the probability that she will select a red marble?

1

Math Prize 2010 Problem 1

If

a

and

b

are nonzero real numbers such that

\left| a \right| \ne \left| b \right|

, compute the value of the expression

\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times \left( \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}} - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}} \right).

2010 Math Prize For Girls Problems

Subcontests

Math Prize 2010 Problem 20

Math Prize 2010 Problem 19

Math Prize 2010 Problem 18

Math Prize 2010 Problem 17

Math Prize 2010 Problem 16

Math Prize 2010 Problem 15

Math Prize 2010 Problem 14

Math Prize 2010 Problem 13

Math Prize 2010 Problem 12

Math Prize 2010 Problem 11

Math Prize 2010 Problem 10

Math Prize 2010 Problem 9

Math Prize 2010 Problem 8

Math Prize 2010 Problem 7

Math Prize 2010 Problem 6

Math Prize 2010 Problem 5

Math Prize 2010 Problem 4

Math Prize 2010 Problem 3

Math Prize 2010 Problem 2

Math Prize 2010 Problem 1