MathDB

2

Octahedron numbering.

The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

Drilling a Tunnel

Let

EFGH

,

EFDC

, and

EHBC

be three adjacent square faces of a cube, for which

EC=8

, and let

A

be the eighth vertex of the cube. Let

I

,

J

, and

K

, be the points on

\overline{EF}

,

\overline{EH}

, and

\overline{EC}

, respectively, so that

EI=EJ=EK=2

. A solid

S

is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to

\overline{AE}

, and containing the edges,

\overline{IJ}

,

\overline{JK}

, and

\overline{KI}

. The surface area of

S

, including the walls of the tunnel, is

m+n\sqrt{p}

, where

m

,

n

, and

p

are positive integers and

p

is not divisible by the square of any prime. Find

m+n+p

.

14

2

Mail Delivery

A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

2n Complex Numbers

There are

2n

complex numbers that satisfy both

z^{28}-z^{8}-1=0

and

|z|=1

. These numbers have the form

z_{m}=\cos\theta_{m}+i\sin\theta_{m}

, where

0\leq\theta_{1}<\theta_{2}< \dots <\theta_{2n}<360

and angles are measured in degrees. Find the value of

\theta_{2}+\theta_{4}+\dots+\theta_{2n}

.

13

2

Arcs

In a certain circle, the chord of a

d

-degree arc is 22 centimeters long, and the chord of a

2d

-degree arc is 20 centimeters longer than the chord of a

3d

-degree arc, where

d<120.

The length of the chord of a

3d

-degree arc is

-m+\sqrt{n}

centimeters, where

m

and

n

are positive integers. Find

m+n.

Chasing Lengths in Quadrilateral

In quadrilateral

ABCD

,

\angle{BAD}\cong\angle{ADC}

and

\angle{ABD}\cong\angle{BCD}

,

AB=8

,

BD=10

, and

BC=6

. The length

CD

may be written in the form

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

12

2

Inscribed Sphere

A sphere is inscribed in the tetrahedron whose vertices are

A=(6,0,0), B=(0,4,0), C=(0,0,2),

and

D=(0,0,0).

The radius of the sphere is

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

Sequence of Polyhedra

Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra

P_{i}

is defined recursively as follows:

P_{0}

is a regular tetrahedron whose volume is 1. To obtain

P_{i+1}

, replace the midpoint triangle of every face of

P_{i}

by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of

P_{3}

is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

11

2

Number, Renumber

In a rectangular array of points, with 5 rows and

N

columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through

N,

the second row is numbered

N+1

through

2N,

and so forth. Five points,

P_1, P_2, P_3, P_4,

and

P_5,

are selected so that each

P_i

is in row

i.

Let

x_i

be the number associated with

P_i.

Now renumber the array consecutively from top to bottom, beginning with the first column. Let

y_i

be the number associated with

P_i

after the renumbering. It is found that

x_1=y_2,

x_2=y_1,

x_3=y_4,

x_4=y_5,

and

x_5=y_3.

Find the smallest possible value of

N.

Club Truncator

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each

\frac{1}{3}

. The probability that Club Truncator will finish the season with more wins than losses is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

10

2

Coordinate Probability

Let

S

be the set of points whose coordinates

x,

y,

and

z

are integers that satisfy

0\le x\le2,

0\le y\le3,

and

0\le z\le4.

Two distinct points are randomly chosen from

S.

The probability that the midpoint of the segment they determine also belongs to

S

is

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

Multiples of 1001

How many positive integer multiples of 1001 can be expressed in the form

10^{j}-10^{i}

, where

i

and

j

are integers and

0\leq i < j \leq 99

?

9

2

Area Ratio

In triangle

ABC

,

AB=13,

BC=15

and

CA=17.

Point

D

is on

\overline{AB},

E

is on

\overline{BC},

and

F

is on

\overline{CA}.

Let

AD=p\cdot AB,

BE=q\cdot BC,

and

CF=r\cdot CA,

where

p,

q,

and

r

are positive and satisfy

p+q+r=2/3

and

p^2+q^2+r^2=2/5.

The ratio of the area of triangle

DEF

to the area of triangle

ABC

can be written in the form

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

Coloring a square grid

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

8

2

7-10 Doubles

Call a positive integer

N

a

<span class='latex-italic'>7-10 double</span>

if the digits of the base-7 representation of

N

form a base-10 number that is twice

N.

For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double?

A Certain Function

A certain function

f

has the properties that

f(3x)=3f(x)

for all positive real values of

x

, and that

f(x)=1-\mid x-2 \mid

for

1\leq x \leq 3

. Find the smallest

x

for which

f(x)=f(2001)

.

7

2

Triangle

ABC

has

AB=21

,

AC=22

, and

BC=20

. Points

D

and

E

are located on

\overline{AB}

and

\overline{AC}

, respectively, such that

\overline{DE}

is parallel to

\overline{BC}

and contains the center of the inscribed circle of triangle

ABC

. Then

DE=m/n

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

Right Triangle and Circles

Let

\triangle{PQR}

be a right triangle with

PQ=90

,

PR=120

, and

QR=150

. Let

C_{1}

be the inscribed circle. Construct

\overline{ST}

with

S

on

\overline{PR}

and

T

on

\overline{QR}

, such that

\overline{ST}

is perpendicular to

\overline{PR}

and tangent to

C_{1}

. Construct

\overline{UV}

with

U

on

\overline{PQ}

and

V

on

\overline{QR}

such that

\overline{UV}

is perpendicular to

\overline{PQ}

and tangent to

C_{1}

. Let

C_{2}

be the inscribed circle of

\triangle{RST}

and

C_{3}

the inscribed circle of

\triangle{QUV}

. The distance between the centers of

C_{2}

and

C_{3}

can be written as

\sqrt{10n}

. What is

n

?

6

2

Rolling Die

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form

m/n,

where

m

and

n

are relatively prime positive integers. Find

m+n.

Squares in Circles

Square

ABCD

is inscribed in a circle. Square

EFGH

has vertices

E

and

F

on

\overline{CD}

and vertices

G

and

H

on the circle. The ratio of the area of square

EFGH

to the area of square

ABCD

can be expressed as

\frac{m}{n}

where

m

and

n

are relatively prime positive integers and

m<n

. Find

10n+m

.

5

2

Triangle inscribed in an ellipse.

An equilateral triangle is inscribed in the ellipse whose equation is

x^2+4y^2=4.

One vertex of the triangle is

(0,1),

one altitude is contained in the

y

-axis, and the length of each side is

\sqrt{\frac mn},

where

m

and

n

are relatively prime positive integers. Find

m+n.

The "Triangle Property"

A set of positive numbers has the

\text{triangle property}

if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets

\{4, 5, 6, \ldots, n\}

of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of

n

?

3

2

Sum of the roots of a crazy binomial.

Find the sum of the roots, real and non-real, of the equation

x^{2001}+\left(\frac 12-x\right)^{2001}=0,

given that there are no multiple roots.

Terms of a Sequence

Given that \begin{align*} x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523, \text{ and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4} \text{ when } n \geq 5, \end{align*}
find the value of

x_{531}+x_{753}+x_{975}

.

4

2

Angle Bisector

In triangle

ABC

, angles

A

and

B

measure 60 degrees and 45 degrees, respectively. The bisector of angle

A

intersects

\overline{BC}

at

T

, and

AT=24.

The area of triangle

ABC

can be written in the form

a+b\sqrt{c},

where

a

,

b

, and

c

are positive integers, and

c

is not divisible by the square of any prime. Find

a+b+c.

Coordinates Problem

Let

R=(8,6)

. The lines whose equations are

8y=15x

and

10y=3x

contain points

P

and

Q

, respectively, such that

R

is the midpoint of

\overline{PQ}

. The length of

PQ

equals

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

2

Averages.

A finite set

\mathcal{S}

of distinct real numbers has the following properties: the mean of

\mathcal{S}\cup\{1\}

is 13 less than the mean of

\mathcal{S}

, and the mean of

\mathcal{S}\cup\{2001\}

is 27 more than the mean of

\mathcal{S}.

Find the mean of

\mathcal{S}.

Students Studying Foreign Languages

Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let

m

be the smallest number of students who could study both languages, and let

M

be the largest number of students who could study both languages. Find

M-m

.

1

2

Find the Largest N

Let

N

be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of

N

forms a perfect square. What are the leftmost three digits of

N

?

Digit Divisibility

Find the sum of all positive two-digit integers that are divisible by each of their digits.

2001 AIME Problems

Subcontests

Octahedron numbering.

Drilling a Tunnel

Mail Delivery

2n Complex Numbers

Arcs

Chasing Lengths in Quadrilateral

Inscribed Sphere

Sequence of Polyhedra

Number, Renumber

Club Truncator

Coordinate Probability

Multiples of 1001

Area Ratio

Coloring a square grid

7-10 Doubles

A Certain Function

Triangle

Right Triangle and Circles

Rolling Die

Squares in Circles

Triangle inscribed in an ellipse.

The "Triangle Property"

Sum of the roots of a crazy binomial.

Terms of a Sequence

Angle Bisector

Coordinates Problem

Averages.

Students Studying Foreign Languages

Find the Largest N

Digit Divisibility

2001 AIME Problems

Subcontests

Octahedron numbering.

Drilling a Tunnel

Mail Delivery

2n Complex Numbers

Arcs

Chasing Lengths in Quadrilateral

Inscribed Sphere

Sequence of Polyhedra

Number, Renumber

Club Truncator

Coordinate Probability

Multiples of 1001

Area Ratio

Coloring a square grid

7-10 Doubles

A Certain Function

Triangle

Right Triangle and Circles

Rolling Die

Squares in Circles

Triangle inscribed in an ellipse.

The &quot;Triangle Property&quot;

Sum of the roots of a crazy binomial.

Terms of a Sequence

Angle Bisector

Coordinates Problem

Averages.

Students Studying Foreign Languages

Find the Largest N

Digit Divisibility

The "Triangle Property"