MathDB

1

counting lattice points under a hyperbola

(x, y)

are there such that

0 < \left\vert xy \right\vert < 36?

2

1

the coefficients of a partial fraction decomposition

If

a

,

b

,

c

,

d

, and

e

are constants such that every

x > 0

satisfies \frac{5x^4 \minus{} 8x^3 \plus{} 2x^2 \plus{} 4x \plus{} 7}{(x \plus{} 2)^4} \equal{} a \plus{} \frac{b}{x \plus{} 2} \plus{} \frac{c}{(x \plus{} 2)^2} \plus{} \frac{d}{(x \plus{} 2)^3} \plus{} \frac{e}{(x \plus{} 2)^4} \, , then what is the value of a \plus{} b \plus{} c \plus{} d \plus{} e?

3

1

recurrence for every third Fibonacci number

The Fibonacci numbers are defined recursively by the equation F_n \equal{} F_{n \minus{} 1} \plus{} F_{n \minus{} 2} for every integer

n \ge 2

, with initial values F_0 \equal{} 0 and F_1 \equal{} 1. Let G_n \equal{} F_{3n} be every third Fibonacci number. There are constants

a

and

b

such that every integer

n \ge 2

satisfies G_n \equal{} a G_{n \minus{} 1} \plus{} b G_{n \minus{} 2}. Compute the ordered pair

(a, b)

.

4

1

admission fees for adults and children

The admission fee for an exhibition is

\$25

per adult and

\$12

per child. Last Tuesday, the exhibition collected

\$1950

in admission fees from at least one adult and at least one child. Of all the possible ratios of adults to children at the exhibition last Tuesday, which one is closest to

1

?

5

1

two tangent circles touching two lines

The figure below shows two parallel lines,

\ell

and

m

, that are distance

12

apart: [asy]unitsize(7); draw((-7, 0) -- (12, 0)); draw((-7, 12) -- (12, 12)); real r = 169 / 48; draw(circle((0, r), r)); draw(circle((5, 12 - r), r)); pair A = (0, 0); pair B = (5, 12); dot(A); dot(B); label("

A

", A, plain.S); label("

B

", B, plain.N); label("

\ell

", (12, 0), plain.E); label("

m

", (12, 12), plain.E);[/asy] A circle is tangent to line

\ell

at point

A

. Another circle is tangent to line

m

at point

B

. The two circles are congruent and tangent to each other as shown. The distance between

A

and

B

is

13

. What is the radius of each circle?

6

1

optional rolling of a die

Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A step starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face up on the die?

8

1

farthest point on a circle

Which point on the circle (x \minus{} 11)^2 \plus{} (y \minus{} 13)^2 \equal{} 116 is farthest from the point

(41, 25)

?

7

1

alternating sum of 4th powers

Compute the value of the expression 2009^4 \minus{} 4 \times 2007^4 \plus{} 6 \times 2005^4 \minus{} 4 \times 2003^4 \plus{} 2001^4 \, .

9

1

counting circles in a grid

The figure below is a

4 \times 4

grid of points. [asy]unitsize(15); for ( int x = 1; x <= 4; ++x ) for ( int y = 1; y <= 4; ++y ) dot((x, y));[/asy]Each pair of horizontally adjacent or vertically adjacent points are distance 1 apart. In the plane of this grid, how many circles of radius 1 pass through exactly two of these grid points?

10

1

dividing a radical power by 9

When the integer {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103} is divided by 9, what is the remainder?

11

1

the 50th term of an arithmetic sequence

An arithmetic sequence consists of

200

numbers that are each at least

10

and at most

100

. The sum of the numbers is

10{,}000

. Let

L

be the least possible value of the

50

th term and let

G

be the greatest possible value of the

50

th term. What is the value of G \minus{} L?

12

1

turning pennies over at random

Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails. She chooses 40 of the pennies at random (all different) and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. At the end, what is the expected number of pennies showing heads?

13

1

an equilateral triangle in a right triangle

The figure below shows a right triangle

\triangle ABC

. [asy]unitsize(15); pair A = (0, 4); pair B = (0, 0); pair C = (4, 0); draw(A -- B -- C -- cycle); pair D = (2, 0); real p = 7 - 3sqrt(3); real q = 4sqrt(3) - 6; pair E = p + (4 - p)*I; pair F = q*I; draw(D -- E -- F -- cycle); label("

A

", A, N); label("

B

", B, S); label("

C

", C, S); label("

D

", D, S); label("

E

", E, NE); label("

F

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

\overline{AB}

and

\overline{BC}

each have length

4

. An equilateral triangle

\triangle DEF

is inscribed in

\triangle ABC

as shown. Point

D

is the midpoint of

\overline{BC}

. What is the area of

\triangle DEF

?

14

1

the powers of the roots of a cubic

The three roots of the cubic 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1 are distinct real numbers between

0

and

1

. For every nonnegative integer

n

, let

s_n

be the sum of the

n

th powers of these three roots. What is the value of the infinite series s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?

15

1

exponential equation (x^x = y^y)

Let x \equal{} \sqrt[3]{\frac{4}{25}}\,. There is a unique value of

y

such that

0 < y < x

and x^x \equal{} y^y. What is the value of

y

? Express your answer in the form

\sqrt[c]{\frac{a}{b}}\,

, where

a

and

b

are relatively prime positive integers and

c

is a prime number.

16

1

five cosines are nonpositive

Let

x

be a real number such that the five numbers

\cos(2 \pi x)

,

\cos(4 \pi x)

,

\cos(8 \pi x)

,

\cos(16 \pi x)

, and

\cos(32 \pi x)

are all nonpositive. What is the smallest possible positive value of

x

?

17

1

three linear equations with a nonlinear objective

Let

a

,

b

,

c

,

x

,

y

, and

z

be real numbers that satisfy the three equations \begin{align*} 13x + by + cz &= 0 \\ ax + 23y + cz &= 0 \\ ax + by + 42z &= 0. \end{align*}Suppose that

a \ne 13

and

x \ne 0

. What is the value of

\frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?

18

1

the digits of 21 factorial

The value of

21!

is

51{,}090{,}942{,}171{,}abc{,}440{,}000

, where

a

,

b

, and

c

are digits. What is the value of 100a \plus{} 10b \plus{} c?

19

1

approximating the diameter of a set of points

Let

S

be a set of

100

points in the plane. The distance between every pair of points in

S

is different, with the largest distance being

30

. Let

A

be one of the points in

S

, let

B

be the point in

S

farthest from

A

, and let

C

be the point in

S

farthest from

B

. Let

d

be the distance between

B

and

C

rounded to the nearest integer. What is the smallest possible value of

d

?

20

1

random cubic polynomial

Let

y_0

be chosen randomly from

\{0, 50\}

, let

y_1

be chosen randomly from

\{40, 60, 80\}

, let

y_2

be chosen randomly from

\{10, 40, 70, 80\}

, and let

y_3

be chosen randomly from

\{10, 30, 40, 70, 90\}

. (In each choice, the possible outcomes are equally likely to occur.) Let

P

be the unique polynomial of degree less than or equal to

3

such that P(0) \equal{} y_0, P(1) \equal{} y_1, P(2) \equal{} y_2, and P(3) \equal{} y_3. What is the expected value of

P(4)

?

2009 Math Prize For Girls Problems

Subcontests

counting lattice points under a hyperbola

the coefficients of a partial fraction decomposition

recurrence for every third Fibonacci number

admission fees for adults and children

two tangent circles touching two lines

optional rolling of a die

farthest point on a circle

alternating sum of 4th powers

counting circles in a grid

dividing a radical power by 9

the 50th term of an arithmetic sequence

turning pennies over at random

an equilateral triangle in a right triangle

the powers of the roots of a cubic

exponential equation (x^x = y^y)

five cosines are nonpositive

three linear equations with a nonlinear objective

the digits of 21 factorial

approximating the diameter of a set of points

random cubic polynomial