MathDB

2

Area of Quadrilateral in Triangle

ABC

is a right angle. Point

D

is on

\overline{BC}

, and

\overline{AD}

bisects angle

CAB

. Points

E

and

F

are on

\overline{AB}

and

\overline{AC}

, respectively, so that

AE=3

and

AF=10.

Given that

EB=9

and

FC=27

, find the integer closest to the area of quadrilateral

DCFG.

[asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label("

A

", A, dir(point--A)); label("

B

", B, dir(point--B)); label("

C

", C, dir(point--C)); label("

D

", D, dir(point--D)); label("

E

", E, dir(point--E)); label("

F

", F, dir(point--F)); label("

G

", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label("10", A--F, dir(90)*dir(A--F)); label("27", F--C, dir(90)*dir(F--C)); label("3", (0,10), W); label("9", (0,4), W);[/asy]

Absent-Minded Professor

While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of

x

for which the sine of

x

degrees is the same as the sine of

x

radians are

\frac{m\pi}{n-\pi}

and

\frac{p\pi}{q+\pi},

where

m,

n,

p

and

q

are positive integers. Find

m+n+p+q.

2

Circle Packing

The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as

\frac{1}{2}\left(\sqrt{p}-q\right),

where

p

and

q

are positive integers. Find

p+q.

[asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); }[/asy]

Surface Area of Cube

Three vertices of a cube are

P=(7,12,10),

Q=(8,8,1),

and

R=(11,3,9).

What is the surface area of the cube?

15

2

Polyhedron

ABCDEFG

has six faces. Face

ABCD

is a square with

AB=12;

face

ABFG

is a trapezoid with

\overline{AB}

parallel to

\overline{GF},

BF=AG=8,

and

GF=6;

and face

CDE

has

CE=DE=14.

The other three faces are

ADEG, BCEF,

and

EFG.

The distance from

E

to face

ABCD

is 12. Given that

EG^2=p-q\sqrt{r},

where

p, q,

and

r

are positive integers and

r

is not divisible by the square of any prime, find

p+q+r.

Intersecting Circles

Circles

\mathcal{C}_{1}

and

\mathcal{C}_{2}

intersect at two points, one of which is

(9,6),

and the product of the radii is

68.

The x-axis and the line

y=mx

, where

m>0,

are tangent to both circles. It is given that

m

can be written in the form

a\sqrt{b}/c,

where

a,

b,

and

c

are positive integers,

b

is not divisible by the square of any prime, and

a

and

c

are relatively prime. Find

a+b+c.

14

2

Sets

A set

\mathcal{S}

of distinct positive integers has the following property: for every integer

x

in

\mathcal{S},

the arithmetic mean of the set of values obtained by deleting

x

from

\mathcal{S}

is an integer. Given that 1 belongs to

\mathcal{S}

and that 2002 is the largest element of

\mathcal{S},

what is the greatet number of elements that

\mathcal{S}

can have?

Right Triangle and Circle

The perimeter of triangle

APM

is

152,

and the angle

PAM

is a right angle. A circle of radius

19

with center

O

on

\overline{AP}

is drawn so that it is tangent to

\overline{AM}

and

\overline{PM}.

Given that

OP=m/n,

where

m

and

n

are relatively prime positive integers, find

m+n.

12

2

Complex function.

Let

F(z)=\frac{z+i}{z-i}

for all complex numbers

z\not= i,

and let

z_n=F(z_{n-1})

for all positive integers

n.

Given that

z_0=\frac 1{137}+i

and

z_{2002}=a+bi,

where

a

and

b

are real numbers, find

a+b.

Basketball Probability

A basketball player has a constant probability of

.4

of making any given shot, independent of previous shots. Let

a_{n}

be the ratio of shots made to shots attempted after

n

shots. The probability that

a_{10}=.4

and

a_{n}\le .4

for all

n

such that

1\le n \le 9

is given to be

p^{a}q^{b}r/(s^{c}),

where

p,

q,

r,

and

s

are primes, and

a,

b,

and

c

are positive integers. Find

(p+q+r+s)(a+b+c).

11

2

Beam of light.

Let

ABCD

and

BCFG

be two faces of a cube with

AB=12.

A beam of light emanates from vertex

A

and reflects off face

BCFG

at point

P,

which is 7 units from

\overline{BG}

and 5 units from

\overline{BC}.

The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point

A

until it next reaches a vertex of the cube is given by

m\sqrt{n},

where

m

and

n

are integers and

n

is not divisible by the square of any prime. Find

m+n.

Two Infinite Geometric Series

Two distinct, real, infinite geometric series each have a sum of

1

and have the same second term. The third term of one of the series is

1/8,

and the second term of both series can be written in the form

\frac{\sqrt{m}-n}{p},

where

m,

n,

and

p

are positive integers and

m

is not divisible by the square of any prime. Find

100m+10n+p.

9

2

Paintable

Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every

h

th picket; Tanya starts with the second picket and paints everth

t

th picket; and Ulysses starts with the third picket and paints every

u

th picket. Call the positive integer

100h+10t+u

<span class='latex-italic'>paintable</span>

when the triple

(h,t,u)

of positive integers results in every picket being painted exaclty once. Find the sum of all the paintable integers.

Counting Disjoint Subsets

Let

\mathcal{S}

be the set

\{1,2,3,\ldots,10\}.

Let

n

be the number of sets of two non-empty disjoint subsets of

\mathcal{S}.

(Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when

n

is divided by

1000.

8

2

Sequence

Find the smallest integer

k

for which the conditions

(1)

a_1, a_2, a_3, \ldots

is a nondecreasing sequence of positive integers

(2)

a_n=a_{n-1}+a_{n-2}

for all

n>2

(3)

a_9=k

are satisfied by more than one sequence.

Floor Function Skipping Integers

Find the least positive integer

k

for which the equation

\lfloor \frac{2002}{n}\rfloor = k

has no integer solutions for

n.

(The notation

\lfloor x \rfloor

means the greatest integer less than or equal to

x.

)

7

2

Binomial fun.

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers

x, y,

and

r

with

|x| > |y|,

(x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots What are the first three digits to the right of the decimal point in the decimal representation of \left(10^{2002} \plus{} 1\right)^{10/7}?

Sum of First k Squares

It is known that, for all positive integers

k,

1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}.

Find the smallest positive integer

k

such that

1^{2}+2^{2}+3^{2}+\cdots+k^{2}

is a multiple of

200.

6

2

System of logs.

The solutions to the system of equations \begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*} are

(x_1,y_1)

and

(x_2, y_2).

Find

\log_{30}{(x_1y_1x_2y_2)}.

Another Frac. Decomposition Problem

Find the integer that is closest to

1000 \sum_{n=3}^{10000}\frac{1}{n^{2}-4}.

5

2

Dodecagon

Let

A_1, A_2, A_3, \ldots, A_{12}

be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set

\{A_1,A_2,A_3,\ldots,A_{12}\}?

Funky Divisors Problem

Find the sum of all positive integers

a=2^{n}3^{m},

where

n

and

m

are non-negative integers, for which

a^{6}

is not a divisor of

6^{a}.

4

2

Sequence

Consider the sequence defined by

a_k=\frac 1{k^2+k}

for

k\ge 1.

Given that

a_m+a_{m+1}+\cdots+a_{n-1}=1/29,

for positive integers

m

and

n

with

m<n

, find

m+n.

Hexagonal Garden

Patio blocks that are hexagons

1

unit on a side are used to outline a garden by placing the blocks edge to edge with

n

on each side. The diagram indicates the path of blocks around the garden when

n=5.

[asy] size(250);int i,j; real r=sqrt(3); for(i=0; i<6; i=i+1) { for(j=0; j<4; j=j+1) { draw(shift(((j*r)*dir(60*i+150)).x, ((j*r)*dir(60*i+150)).y)*shift((4r*dir(60i+30)).x,(4r*dir(60i+30)).y)*polygon(6)); }}[/asy] If

n=202,

then the area of the garden enclosed by the path, not including the path itself, is

m(\sqrt{3}/2)

square units, where

m

is a positive integer. Find the remainder when

m

is divided by

1000.

3

2

Dick and Jane

Jane is 25 years old. Dick is older than Jane. In

n

years, where

n

is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let

d

be Dick's present age. How many ordered pairs of positive integers

(d,n)

are possible?

Geometric Series & Logarithms

It is given that

\log_{6}a+\log_{6}b+\log_{6}c=6,

where

a,

b,

and

c

are positive integers that form an increasing geometric sequence and

b-a

is the square of an integer. Find

a+b+c.

1

2

License plate

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is

m/n

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

Counting z=abc-cba

Given that \begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of

z

are possible?

13

2

Intermediate Forum AIME Marathon

In triangle

ABC

the medians

\overline{AD}

and

\overline{CE}

have lengths 18 and 27, respectively, and AB \equal{} 24. Extend

\overline{CE}

to intersect the circumcircle of

ABC

at

F

. The area of triangle

AFB

is

m\sqrt {n}

, where

m

and

n

are positive integers and

n

is not divisible by the square of any prime. Find m \plus{} n.

Area Ratio in Triangle

In triangle

ABC,

point

D

is on

\overline{BC}

with

CD=2

and

DB=5,

point

E

is on

\overline{AC}

with

CE=1

and

EA=3,

AB=8,

and

\overline{AD}

and

\overline{BE}

intersect at

P.

Points

Q

and

R

lie on

\overline{AB}

so that

\overline{PQ}

is parallel to

\overline{CA}

and

\overline{PR}

is parallel to

\overline{CB}.

It is given that the ratio of the area of triangle

PQR

to the area of triangle

ABC

is

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

2002 AIME Problems

Subcontests

Area of Quadrilateral in Triangle

Absent-Minded Professor

Circle Packing

Surface Area of Cube

Polyhedron

Intersecting Circles

Sets

Right Triangle and Circle

Complex function.

Basketball Probability

Beam of light.

Two Infinite Geometric Series

Paintable

Counting Disjoint Subsets

Sequence

Floor Function Skipping Integers

Binomial fun.

Sum of First k Squares

System of logs.

Another Frac. Decomposition Problem

Dodecagon

Funky Divisors Problem

Sequence

Hexagonal Garden

Dick and Jane

Geometric Series & Logarithms

License plate

Counting z=abc-cba

Intermediate Forum AIME Marathon

Area Ratio in Triangle

2002 AIME Problems

Subcontests

Area of Quadrilateral in Triangle

Absent-Minded Professor

Circle Packing

Surface Area of Cube

Polyhedron

Intersecting Circles

Sets

Right Triangle and Circle

Complex function.

Basketball Probability

Beam of light.

Two Infinite Geometric Series

Paintable

Counting Disjoint Subsets

Sequence

Floor Function Skipping Integers

Binomial fun.

Sum of First k Squares

System of logs.

Another Frac. Decomposition Problem

Dodecagon

Funky Divisors Problem

Sequence

Hexagonal Garden

Dick and Jane

Geometric Series &amp; Logarithms

License plate

Counting z=abc-cba

Intermediate Forum AIME Marathon

Area Ratio in Triangle

Geometric Series & Logarithms