MathDB

2

Stack of 2000 Cards

2000

cards is labelled with the integers from

1

to

2000,

with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order:

1,2,3,\ldots,1999,2000.

In the original stack of cards, how many cards were above the card labelled 1999?

Trig Summation

Find the least positive integer

n

such that

\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.

14

2

Angles in Triangle

In triangle

ABC,

it is given that angles

B

and

C

are congruent. Points

P

and

Q

lie on

\overline{AC}

and

\overline{AB},

respectively, so that

AP=PQ=QB=BC.

Angle

ACB

is

r

times as large as angle

APQ,

where

r

is a positive real number. Find the greatest integer that does not exceed

1000r.

Factorial Base Expansions

Every positive integer

k

has a unique factorial base expansion

(f_1,f_2,f_3,\ldots,f_m),

meaning that

k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m,

where each

f_i

is an integer,

0\le f_i\le i,

and

0<f_m.

Given that

(f_1,f_2,f_3,\ldots,f_j)

is the factorial base expansion of

16!-32!+48!-64!+\cdots+1968!-1984!+2000!,

find the value of

f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j.

13

2

Firetruck and Area Covered

In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at

50

miles per hour along the highways and at

14

miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is

m/n

square miles, where

m

and

n

are relatively prime positive integers. Find

m+n.

Equation Roots

The equation

2000x^6+100x^5+10x^3+x-2=0

has exactly two real roots, one of which is

\frac{m+\sqrt{n}}r,

where

m, n

and

r

are integers,

m

and

r

are relatively prime, and

r>0.

Find

m+n+r.

12

2

Different Values of Function

Given a function

f

for which

f(x)=f(398-x)=f(2158-x)=f(3214-x)

holds for all real

x,

what is the largest number of different values that can appear in the list

f(0),f(1),f(2),\ldots,f(999)?

Sphere

The points

A, B

and

C

lie on the surface of a sphere with center

O

and radius 20. It is given that

AB=13, BC=14, CA=15,

and that the distance from

O

to triangle

ABC

is

\frac{m\sqrt{n}}k,

where

m, n,

and

k

are positive integers,

m

and

k

are relatively prime, and

n

is not divisible by the square of any prime. Find

m+n+k.

11

2

Manipulating Divisors

Let

S

be the sum of all numbers of the form

a/b,

where

a

and

b

are relatively prime positive divisors of

1000.

What is the greatest integer that does not exceed

S/10?

Analytic Geometry

The coordinates of the vertices of isosceles trapezoid

ABCD

are all integers, with

A=(20,100)

and

D=(21,107).

The trapezoid has no horizontal or vertical sides, and

\overline{AB}

and

\overline{CD}

are the only parallel sides. The sum o f the absolute values of all possible slopes for

\overline{AB}

is

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

10

2

Sequence of Numbers

A sequence of numbers

x_{1},x_{2},x_{3},\ldots,x_{100}

has the property that, for every integer

k

between

1

and

100,

inclusive, the number

x_{k}

is

k

less than the sum of the other

99

numbers. Given that

x_{50}=m/n,

where

m

and

n

are relatively prime positive integers, find

m+n.

Inscribed Circle

A circle is inscribed in quadrilateral

ABCD,

tangent to

\overline{AB}

at

P

and to

\overline{CD}

at

Q.

Given that

AP=19, PB=26, CQ=37,

and

QD=23,

find the square of the radius of the circle.

9

2

System of Equations: Logs

The system of equations \begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*} has two solutions

(x_{1},y_{1},z_{1})

and

(x_{2},y_{2},z_{2}).

Find

y_{1} + y_{2}.

Classic Complex Numbers

Given that

z

is a complex number such that

z+\frac 1z=2\cos 3^\circ,

find the least integer that is greater than

z^{2000}+\frac 1{z^{2000}}.

8

2

Classic Cone Problem

A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is

m-n\sqrt[3]{p},

where

m,

n,

and

p

are positive integers and

p

is not divisible by the cube of any prime number. Find

m+n+p.

Trapezoid

In trapezoid

ABCD,

leg

\overline{BC}

is perpendicular to bases

\overline{AB}

and

\overline{CD},

and diagonals

\overline{AC}

and

\overline{BD}

are perpendicular. Given that

AB=\sqrt{11}

and

AD=\sqrt{1001},

find

BC^2.

7

2

Factorials Reciprocals

Given that

\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}

find the greatest integer that is less than

\frac N{100}.

System of Equations

Suppose that

x,

y,

and

z

are three positive numbers that satisfy the equations

xyz=1,

x+\frac{1}{z}=5,

and

y+\frac{1}{x}=29.

Then

z+\frac{1}{y}=\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m+n.

6

2

AM is two more than GM

For how many ordered pairs

(x,y)

of integers is it true that

0<x<y<10^{6}

and that the arithmetic mean of

x

and

y

is exactly

2

more than the geometric mean of

x

and

y?

Trapezoid Area

One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio

2: 3.

Let

x

be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed

x^2/100.

5

2

Black and White Marbles

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is

25.

One marble is taken out of each box randomly. The probability that both marbles are black is

27/50,

and the probability that both marbles are white is

m/n,

where

m

and

n

are relatively prime positive integers. What is

m+n?

Ring Arrangements

Given eight distinguishable rings, let

n

be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of

n.

4

2

Dissected Rectangle

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Even and Odd Divisors

What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?

3

2

Binomial Expansion

In the expansion of

(ax+b)^{2000},

where

a

and

b

are relatively prime positive integers, the coefficients of

x^{2}

and

x^{3}

are equal. Find

a+b.

Card Probability

A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let

m/n

be the probability that two randomly selected cards also form a pair, where

m

and

n

are relatively prime positive integers. Find

m+n.

2

Point Reflections

Let

u

and

v

be integers satisfying

0<v<u.

Let

A=(u,v),

let

B

be the reflection of

A

across the line

y=x,

let

C

be the reflection of

B

across the y-axis, let

D

be the reflection of

C

across the x-axis, and let

E

be the reflection of

D

across the y-axis. The area of pentagon

ABCDE

is 451. Find

u+v.

Hyperbola Lattice Points

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola

x^2-y^2=2000^2.

1

2

Powers of 10

Find the least positive integer

n

such that no matter how

10^{n}

is expressed as the product of any two positive integers, at least one of these two integers contains the digit

0.

Simple Logarithms

The number

\frac 2{\log_4{2000^6}}+\frac 3{\log_5{2000^6}}

can be written as

\frac mn

where

m

and

n

are relatively prime positive integers. Find

m+n.

2000 AIME Problems

Subcontests

Stack of 2000 Cards

Trig Summation

Angles in Triangle

Factorial Base Expansions

Firetruck and Area Covered

Equation Roots

Different Values of Function

Sphere

Manipulating Divisors

Analytic Geometry

Sequence of Numbers

Inscribed Circle

System of Equations: Logs

Classic Complex Numbers

Classic Cone Problem

Trapezoid

Factorials Reciprocals

System of Equations

AM is two more than GM

Trapezoid Area

Black and White Marbles

Ring Arrangements

Dissected Rectangle

Even and Odd Divisors

Binomial Expansion

Card Probability

Point Reflections

Hyperbola Lattice Points

Powers of 10

Simple Logarithms