MathDB

1

Another System

a

,

b

,

c

be nonzero real numbers such that a\plus{}b\plus{}c\equal{}0 and a^3\plus{}b^3\plus{}c^3\equal{}a^5\plus{}b^5\plus{}c^5. Find the value of a^2\plus{}b^2\plus{}c^2.

32

1

Another Cyclic Pentagon

Cyclic pentagon

ABCDE

has side lengths AB\equal{}BC\equal{}5, CD\equal{}DE\equal{}12, and AE \equal{} 14. Determine the radius of its circumcircle.

31

1

Sequence of Hyperbola Points

Let

\mathcal{C}

be the hyperbola y^2 \minus{} x^2 \equal{} 1. Given a point

P_0

on the

x

-axis, we construct a sequence of points

(P_n)

on the

x

-axis in the following manner: let

\ell_n

be the line with slope

1

passing passing through

P_n

, then P_{n\plus{}1} is the orthogonal projection of the point of intersection of

\ell_n

and

\mathcal C

onto the

x

-axis. (If P_n \equal{} 0, then the sequence simply terminates.) Let

N

be the number of starting positions

P_0

on the

x

-axis such that P_0 \equal{} P_{2008}. Determine the remainder of

N

when divided by

2008

.

30

1

Triangle Side System

Triangle

ABC

obeys

AB = 2AC

and

\angle{BAC} = 120^{\circ}.

Points

P

and

Q

lie on segment

BC

such that \begin{eqnarray*} AB^2 + BC \cdot CP = BC^2 \\ 3AC^2 + 2BC \cdot CQ = BC^2 \end{eqnarray*} Find

\angle{PAQ}

in degrees.

29

1

System

Let

(x,y)

be a pair of real numbers satisfying 56x \plus{} 33y \equal{} \frac{\minus{}y}{x^2\plus{}y^2}, \qquad \text{and} \qquad 33x\minus{}56y \equal{} \frac{x}{x^2\plus{}y^2}. Determine the value of |x| \plus{} |y|.

28

1

Polyhedron

Let

P

be a polyhedron where every face is a regular polygon, and every edge has length

1

. Each vertex of

P

is incident to two regular hexagons and one square. Choose a vertex

V

of the polyhedron. Find the volume of the set of all points contained in

P

that are closer to

V

than to any other vertex.

27

1

Cyclic Pentagon

Cyclic pentagon

ABCDE

has a right angle \angle{ABC} \equal{} 90^{\circ} and side lengths AB \equal{} 15 and BC \equal{} 20. Supposing that AB \equal{} DE \equal{} EA, find

CD

.

26

1

Parabola

Let

\mathcal P

be a parabola, and let

V_1

and

F_1

be its vertex and focus, respectively. Let

A

and

B

be points on

\mathcal P

so that \angle AV_1 B \equal{} 90^\circ. Let

\mathcal Q

be the locus of the midpoint of

AB

. It turns out that

\mathcal Q

is also a parabola, and let

V_2

and

F_2

denote its vertex and focus, respectively. Determine the ratio

F_1F_2/V_1V_2

.

25

1

Alice and the Cheshire Cat

Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from

n

to

5n

or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (and the cat disappears) as soon as the number of beans Alice has is greater than

2008

and has last two digits

42

. What is the minimum number of cents Alice can spend to win the game, assuming she starts with 0 beans?

24

1

Isosceles Trisection

Suppose that

ABC

is an isosceles triangle with AB \equal{} AC. Let

P

be the point on side

AC

so that AP \equal{} 2CP. Given that BP \equal{} 1, determine the maximum possible area of

ABC

.

23

1

Kelly + Jason

Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer

n < 60

(both Kelly and Jason know that

n < 60

, but that they don't know what the value of

n

is). The computer tells Kelly the unit digit of

n

, and it tells Jason the number of divisors of

n

. Then, Kelly and Jason have the following dialogue: Kelly: I don't know what

n

is, and I'm sure that you don't know either. However, I know that

n

is divisible by at least two different primes. Jason: Oh, then I know what the value of

n

is. Kelly: Now I also know what

n

is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of

n

?

22

1

Theta Function

For a positive integer

n

, let

\theta(n)

denote the number of integers

0 \leq x < 2010

such that x^2 \minus{} n is divisible by

2010

. Determine the remainder when \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n) is divided by

2010

.

18

1

Geometric Ratio

Let

ABC

be a right triangle with \angle A \equal{} 90^\circ. Let

D

be the midpoint of

AB

and let

E

be a point on segment

AC

such that AD \equal{} AE. Let

BE

meet

CD

at

F

. If \angle BFC \equal{} 135^\circ, determine

BC / AB

.

17

1

Many Square Roots

Solve the equation \sqrt {x \plus{} \sqrt {4x \plus{} \sqrt {16x \plus{} \sqrt {\dotsc \plus{} \sqrt {4^{2008}x \plus{} 3}}}}} \minus{} \sqrt {x} \equal{} 1. Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.

16

1

Angle Maximization

Point

A

lies at

(0, 4)

and point

B

lies at

(3, 8)

. Find the

x

-coordinate of the point

X

on the

x

-axis maximizing

\angle AXB

.

15

1

Game Show

In a game show, Bob is faced with

7

doors,

2

of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize?

14

1

Infinite Sum

Evaluate the infinite sum \sum_{n\equal{}1}^{\infty}\frac{n}{n^4\plus{}4}.

13

1

Polynomial and Values

Let

P(x)

be a polynomial with degree 2008 and leading coefficient 1 such that P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0. Determine the value of

P(2008)

. You may use factorials in your answer.

21

1

Square Minimization

Let

ABC

be a triangle with AB \equal{} 5, BC \equal{} 4 and AC \equal{} 3. Let

\mathcal P

and

\mathcal Q

be squares inside

ABC

with disjoint interiors such that they both have one side lying on

AB

. Also, the two squares each have an edge lying on a common line perpendicular to

AB

, and

\mathcal P

has one vertex on

AC

and

\mathcal Q

has one vertex on

BC

. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph;
unitsize(1.5cm);
pair A, B, C;
A = origin; B = A + 5 * right; C = (9/5, 12/5);
pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45);
pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C);
pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2;
pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y);
draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle);
label("

A

", A, W); label("

B

", B, E); label("

C

", C, N);
label("

\mathcal P

", Oa, origin); label("

\mathcal Q

", Ob, origin);[/asy]

20

1

Symmetric Diophantine

For how many ordered triples

(a,b,c)

of positive integers are the equations abc\plus{}9 \equal{} ab\plus{}bc\plus{}ca and a\plus{}b\plus{}c \equal{} 10 satisfied?

19

1

Tetrahedral Minimization

Let

ABCD

be a regular tetrahedron, and let

O

be the centroid of triangle

BCD

. Consider the point

P

on

AO

such that

P

minimizes PA \plus{} 2(PB \plus{} PC \plus{} PD). Find

\sin \angle PBO

.

12

1

Conic Section + Inscribed Sphere

Suppose we have an (infinite) cone

\mathcal C

with apex

A

and a plane

\pi

. The intersection of

\pi

and

\mathcal C

is an ellipse

\mathcal E

with major axis

BC

, such that

B

is closer to

A

than

C

, and BC \equal{} 4, AC \equal{} 5, AB \equal{} 3. Suppose we inscribe a sphere in each part of

\mathcal C

cut up by

\mathcal E

with both spheres tangent to

\mathcal E

. What is the ratio of the radii of the spheres (smaller to larger)?

11

1

Infinite Sum

Let f(r) \equal{} \sum_{j \equal{} 2}^{2008} \frac {1}{j^r} \equal{} \frac {1}{2^r} \plus{} \frac {1}{3^r} \plus{} \dots \plus{} \frac {1}{2008^r}. Find \sum_{k \equal{} 2}^{\infty} f(k).

5

5

5

6

5

6

6

6

7

2008 Harvard-MIT Mathematics Tournament

Subcontests

Another System

Another Cyclic Pentagon

Sequence of Hyperbola Points

Triangle Side System

System

Polyhedron

Cyclic Pentagon

Parabola

Alice and the Cheshire Cat

Isosceles Trisection

Kelly + Jason

Theta Function

Geometric Ratio

Many Square Roots

Angle Maximization

Game Show

Infinite Sum

Polynomial and Values

Square Minimization

Symmetric Diophantine

Tetrahedral Minimization

Conic Section + Inscribed Sphere

Infinite Sum