KZ-MO 2004

For reals $1\leq a\leq b \leq c \leq d \leq e \leq f$ prove inequality $(af + be + cd)(af + bd + ce) \leq (a + b^2 + c^3 )(d + e^2 + f^3 )$ .

A zigzag is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $n$ zigzags?

Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

In some village there are $1000$ inhabitants. Every day, each of them shares the news they learned yesterday with all their friends. It is known that any news becomes known to all residents of the village. Prove that it is possible to select $90$ residents so that if you tell all of them at the same time some news, then in $10$ days it will become known to all residents of the village.

Let $P (x)$ be a polynomial with real coefficients such that $P (x)> 0$ for all $x \geq 0$ . Prove that there is a positive integer $n$ such that $(1 + x) ^ n P (x)$ polynomial with nonnegative coefficients.

The sequence of integers $a_1$ , $a_2$ , $\dots$ is defined as follows: $a_1 = 1$ and $n> 1$ , $a_ {n + 1}$ is the smallest integer greater than $a_n$ and such, that $a_i + a_j \neq 3a_k$ for any $i, j$ and $k$ from $\{1, 2, \dots, n + 1 \}$ are not necessarily different. Define $a_ {2004}$ .

Prove that for any $a>0,b>0,c>0$ we have $8a^2 b^2 c^2 \geq (a^2 + ab + ac - bc)(b^2 + ba + bc - ac)(c^2 + ca + cb - ab)$ .