KG-MO 2010

Given that $a,b,c > 0$ and $a + b + c = 1$ . Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$ .

Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.

At the meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each with one another, among the remaining people they do not have a common friend. For each pair individuals $A$ and $B$ are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?

Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$ , respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.

Let $k$ be a constant number larger than $1$ . Find all polynomials $P(x)$ such that $P({x^k}) = {\left( {P(x)} \right)^k}$ for all real $x$ .

Let $p$ - a prime, where $p> 11$ . Prove that there exists a number $k$ such that the product $p \cdot k$ can be written in the decimal system with only ones.

Find all natural triples $(a,b,c)$ , such that: $a - ) a \le b \le c$ $b - )\,(a,b,c) = 1$ $c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$ .

Solve in none-negative integers ${x^3} + 7{x^2} + 35x + 27 = {y^3}$ .