RS-SMO 2012

Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$ . Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$ . Prove that $$\angle AED=\angle PEB.$$

Find all natural numbers $a$ and $b$ such that $$a|b^2, b|a^2 \text{ and } a+1|b^2+1.$$

Find all natural numbers $n$ for which there is a permutation $(p_1,p_2,...,p_n)$ of numbers $(1,2,...,n)$ such that sets $\{p_1 +1, p_2 + 2,..., p_n +n\}$ and $\{p_1-1, p_2-2,...,p_n -n\}$ are complete residue systems $\mod n$ .

Let $\mathbb{K}$ be two-dimensional integer lattice. Is there a bijection $f:\mathbb{N} \rightarrow \mathbb{K}$ , such that for every distinct $a,b,c \in \mathbb{N}$ we have: $$\gcd(a,b,c)>1 \Rightarrow f(a),f(b),f(c) \text{ are not colinear? }$$