1990 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

2

Tournament and diving into groups

In a tournament, every two of the

n

players played exactly one match with each other (no draws). Prove that it is possible either (i) to partition the league in two groups

A

B

such that everybody in

A

defeated everybody in

B

; or (ii) to arrange all the players in a chain

x_1, x_2, . . . , x_n, x_1

in such a way that each player defeated his successor.

Sum divisible by 3

Prove that for all integers

n > 2

3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i}

2

cyclic inequality with fractions

x_1, x_2, . . . , x_n

be positive numbers. Prove that

\sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1

x_{n+1}=x_1

x_{n+2}=x_2

Sequence and perfect squares

(a_n)

is a sequence of positive integers such that

\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0

Prove that there exists

k

such that there are at least

1990

perfect squares between

a_1 + a_2 + ... + a_k

a_1 + a_2 + ... + a_{k+1}

2

functional equation in R

Find all functions

f : \mathbb{R} \longrightarrow \mathbb{R}

(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)

Triangle inscribed in a square

A triangle whose all sides have length not smaller than

1

is inscribed in a square of side length

1

. Prove that the center of the square lies inside the triangle or on its boundary.