BG-MO 2018

Let n be an odd positive integer.Let M be a set of $n$ positive integers ,which are 2x2 different. A set $T$ $\in$ $M$ is called "good" if the product of its elements is divisible by the sum of the elements in $M$ , but is not divisible by the square of the same sum. Given that $M$ is "good",how many "good" subsets of $M$ can there be?

Let $ABCD$ be a cyclic quadrilateral. Let $H_{1}$ be the orthocentre of triangle $ABC$ . Point $A_{1}$ is the image of $A$ after reflection about $BH_{1}$ . Point $B_{1}$ is the image of of $B$ after reflection about $AH_{1}$ . Let $O_{1}$ be the circumcentre of $(A_{1}B_{1}H_{1})$ . Let $H_{2}$ be the orthocentre of triangle $ABD$ . Point $A_{2}$ is the image of $A$ after reflection about $BH_{2}$ . Point $B_{2}$ is the image of of $B$ after reflection about $AH_{2}$ . Let $O_{2}$ be the circumcentre of $(A_{2}B_{2}H_{2})$ . Lets denote by $\ell_{AB}$ be the line through $O_{1}$ and $O_{2}$ . $\ell_{AD}$ , $\ell_{BC}$ , $\ell_{CD}$ are defined analogously. Let $M=\ell_{AB} \cap \ell_{BC}$ , $N=\ell_{BC} \cap \ell_{CD}$ , $P=\ell_{CD} \cap \ell_{AD}$ , $Q=\ell_{AD} \cap \ell_{AB}$ . Prove that $MNPQ$ is cyclic.

Prove that $$\left(\frac{6}{5}\right)^{\sqrt{3}}>\left(\frac{5}{4}\right)^{\sqrt{2}}.$$

Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$, $I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.

Given a polynomial $P(x)=a_{d}x^{d}+ \ldots +a_{2}x^{2}+a_{0}$ with positive integers for coefficients and degree $d\geq 2$ . Consider the sequence defined by $$b_{1}=a_{0} ,b_{n+1}=P(b_{n})$$ for $n \geq 1$ . Prove that for all $n \geq 2$ there exists a prime $p$ such that $p$ divides $b_{n}$ but does not divide $b_{1}b_{2} \ldots b_{n-1}$ .

On a planet there are $M$ countries and $N$ cities. There are two-way roads between some of the cities. It is given that:

(1) In each county there are at least three cities;

(2) For each country and each city in the country is connected by roads with at least half of the other cities in the countries;

(3) Each city is connceted with exactly one other city ,that is not in its country;

(4) There are at most two roads between cities from cities in two different countries;

(5) If two countries contain less than $2M$ cities in total then there is a road between them.

Prove that there is cycle of lenght at least $M+\frac{N}{2}$ .