IR-MO-R2 2017

a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$ b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$ . Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$ . Prove that $$AB+CD>DA+BC.$$

Let $n$ be a natural number divisible by $3$ . We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ( $m > 1$ ), the number of black squares is not more than white squares. Find the maximum number of black squares.

Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$ . Prove that $$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$