MathDB

1

Two congruent cubes interlocked

Find out whether there exist two congruent cubes with a common center such that each face of one cube has a common point with each face of the other.

11

1

Skiing competition with one change of leader

In a skiing jump competition

65

contestants take part. They jump with the previously established order. Each of them jumps once. We assume that the obtained results are different and the orders of the contestants after the competition are equally likely. In each moment of the competition by a leader we call a person who is scored the best at this moment. Denote by

p

the probability that during the whole competition there was exactly one change of the leader. Prove that

p > 1/16

.

10

1

x^x = y^3 + z^3 diophantine

Prove that the equation

x^x = y^3+z^3

has infinitely many solutions in positive integers

x,y,z

.

9

1

Modular polynomial arithmetic, polynomial has no integer roots

A polynomial with integer coefficients when divided by

x^2-12x+11

gives the remainder

990x-889

. Prove that the polynomial has no integer roots.

8

1

Bouncing ray of light

The ray of light starts from the center of a square and reflects from its sides with the principle that the angle of reflection is equal to the angle of incidence. After some time the ray returns to the center of the square. The ray never reached the vertex and has never returned to the center of the square before. Prove that the ray reflected from the sides of the square an odd number of times.

7

1

Inequality on similar looking numbers (a+b+c = p+q+r = 1)

Nonnegative numbers

a, b, c, p, q, r

satisfy the conditions:

a + b + c = p + q + r = 1; ~~~~~~ p, q, r \leq \frac{1}{2}

. Prove that

8abc \leq pa + qb + rc

and determine when equality holds.

6

1

Arithmetic and geometric sequence with one term in common

Given two sequences of positive integers: the arithmetic sequence with difference

r > 0

and the geometric sequence with ratio

q > 1

;

r

and

q

are coprime. Prove that if these sequences have one term in common, then they have them infinitely many.

5

1

Inequality involving the tangent of an angle

Given triangle

ABC

in the plane such that

\angle CAB = a > \pi/2

. Let

PQ

be a segment whose midpoint is the point

A

. Prove that

(BP+CQ) \tan a/2 \geq BC

.

4

1

Line tangent to an incircle of a triangle

A line tangent to the incircle of the equilateral triangle ABC intersects the sides AB and BC at points D and E respectively. Prove that

\frac{AD}{DB}+\frac{AE}{EC} = 1

.

3

1

Picking people from a group that know each other

In a group of

kn

persons, each person knows more than

(k-1)n

others (

k,n

are positive integers). Prove that one can choose

k+1

persons from this group so that each chosen person knows all the others chosen.
Note: If a person

A

knows

B

, then

B

knows

A

.

2

1

On primes which are divisors of a difference of consecutive palindromes

A number is called a palindromic number if its decimal representation read from the left to the right is the same as read from the right to the left. Let

(x_n)

be the increasing sequence of all palindromic numbers. Determine all primes, which are divisors of at least one of the differences

x_{n+1} - x_n

.

1

2sin(nx) = tan(x) + cot(x)

Determine all positive integers

n

, such that the equation

2 \sin nx = \tan x + \cot x

has solutions in real numbers

x

.

1995 Poland - First Round

Subcontests

Two congruent cubes interlocked

Skiing competition with one change of leader

x^x = y^3 + z^3 diophantine

Modular polynomial arithmetic, polynomial has no integer roots

Bouncing ray of light

Inequality on similar looking numbers (a+b+c = p+q+r = 1)

Arithmetic and geometric sequence with one term in common

Inequality involving the tangent of an angle

Line tangent to an incircle of a triangle

Picking people from a group that know each other

On primes which are divisors of a difference of consecutive palindromes

2sin(nx) = tan(x) + cot(x)