MathDB

2

Tournament Points

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

Square to Polyhedron

Three 12 cm

\times

12 cm squares are each cut into two pieces

A

and

B

, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in

\text{cm}^3

) of this polyhedron?
[asy] defaultpen(fontsize(10)); size(250); draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4)); draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4)); draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4)); filldraw(scale(2)*polygon(6), white, black); pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225); pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)}; draw(roots[0]--X--roots[1]); label("

B

", centroid(roots[0],X,roots[1])); draw(roots[2]--Y--roots[3]); label("

B

", centroid(roots[2],Y,roots[3])); draw(roots[4]--Z--roots[5]); label("

B

", centroid(roots[4],Z,roots[5])); label("

A

", (1+sqrt(3))*dir(90)); label("

A

", (1+sqrt(3))*dir(210)); label("

A

", (1+sqrt(3))*dir(330)); draw(shift(-10,0)*scale(2)*polygon(4)); draw((sqrt(2)-10,0)--(-10,sqrt(2))); label("

A

", (-10,0)); label("

B

", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]

13

1

GCD of consecutive terms

The numbers in the sequence 101, 104, 109, 116,

\dots

are of the form

a_n = 100 + n^2

, where

n = 1

, 2, 3,

\dots

. For each

n

, let

d_n

be the greatest common divisor of

a_n

and

a_{n + 1}

. Find the maximum value of

d_n

as

n

ranges through the positive integers.

10

1

A representation using floor

How many of the first 1000 positive integers can be expressed in the form

\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor,

where

x

is a real number, and

\lfloor z \rfloor

denotes the greatest integer less than or equal to

z

?

9

1

Parallel chords

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of

\alpha

,

\beta

, and

\alpha + \beta

radians, respectively, where

\alpha + \beta < \pi

. If

\cos \alpha

, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

7

1

Four integer variables

Assume that

a

,

b

,

c

, and

d

are positive integers such that

a^5 = b^4

,

c^3 = d^2

, and

c - a = 19

. Determine

d - b

.

6

1

A triangle with six parts

As shown in the figure, triangle

ABC

is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle

ABC

.
[asy] size(200); pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C); draw(F--C--A--B--C^^A--D^^B--E); label("

A

", A, SW); label("

B

", B, SE); label("

C

", C, N); label("84", centroid(H, C, E), fontsize(9.5)); label("35", centroid(H, B, D), fontsize(9.5)); label("30", centroid(H, F, B), fontsize(9.5)); label("40", centroid(H, A, F), fontsize(9.5));[/asy]

5

1

Partial sums of a recurrent sequence

A sequence of integers

a_1

,

a_2

,

a_3

,

\ldots

is chosen so that

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

for each

n \ge 3

. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

8

1

Integer Approximations

The sum of the following seven numbers is exactly 19:

a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,

a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.

It is desired to replace each

a_i

by an integer approximation

A_i

,

1 \le i \le 7

, so that the sum of the

A_i

's is also 19 and so that

M

, the maximum of the "errors"

|A_i - a_i|

, is as small as possible. For this minimum

M

, what is

100M

?

4

1

Small Square

A small square is constructed inside a square of area 1 by dividing each side of the unit square into

n

equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of

n

if the the area of the small square is exactly 1/1985. [asy] size(200); pair A=(0,1), B=(1,1), C=(1,0), D=origin; draw(A--B--C--D--A--(1,1/6)); draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1)); pair point=( 0.5 , 0.5 ); //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("

1/n

", (11/12,1), N, fontsize(9));[/asy]

3

1

Some imaginary numbers

Find

c

if

a

,

b

, and

c

are positive integers which satisfy

c=(a + bi)^3 - 107i

, where

i^2 = -1

.

2

1

Rotating a Right Triangle

When a right triangle is rotated about one leg, the volume of the cone produced is

800 \pi

\text{cm}^3

. When the triangle is rotated about the other leg, the volume of the cone produced is

1920 \pi

\text{cm}^3

. What is the length (in cm) of the hypotenuse of the triangle?

1

Fun Product

Let

x_1 = 97

, and for

n > 1

let

x_n = \frac{n}{x_{n - 1}}

. Calculate the product

x_1 x_2 \dotsm x_8

.

11

1

Ellipse not parallel to x- or y-axis

An ellipse has foci at

(9,20)

and

(49,55)

in the

xy

-plane and is tangent to the

x

-axis. What is the length of its major axis?

12

1

AIME Problem

Let

A

,

B

,

C

, and

D

be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex

A

, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let

p = n/729

be the probability that the bug is at vertex

A

when it has crawled exactly 7 meters. Find the value of

n

.

1985 ITAMO

Subcontests

Tournament Points

Square to Polyhedron

GCD of consecutive terms

A representation using floor

Parallel chords

Four integer variables

A triangle with six parts

Partial sums of a recurrent sequence

Integer Approximations

Small Square

Some imaginary numbers

Rotating a Right Triangle

Fun Product

Ellipse not parallel to x- or y-axis

AIME Problem