SILK-ROAD 2009

Prove that, abc≤1 and a,b,c>0 $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c}$$

Bisectors of triangle ABC of an angles A and C intersect with BC and AB at points A1 and C1 respectively. Lines AA1 and CC1 intersect circumcircle of triangle ABC at points A2 and C2 respectively. K is intersection point of C1A2 and A1C2. I is incenter of ABC. Prove that the line KI divides AC into two equal parts.

A tourist going to visit the Complant, found that: a) in this country $1024$ cities, numbered by integers from $0$ to $1023$ , b) two cities with numbers $m$ and $n$ are connected by a straight line if and only if the binary entries of numbers $m$ and $n$ they differ exactly in one digit, c) during the stay of a tourist in that country $8$ roads will be closed for scheduled repairs. Prove that a tourist can make a closed route along the existing roads of Complant, passing through each of its cities exactly once.

Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.