OMM 2013

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$ , such that $p_a - p_b$ divides $2(a-b)$ .

Let $ABCD$ be a parallelogram with the angle at $A$ obtuse. Let $P$ be a point on segment $BD$ . The circle with center $P$ passing through $A$ cuts line $AD$ at $A$ and $Y$ and cuts line $AB$ at $A$ and $X$ . Line $AP$ intersects $BC$ at $Q$ and $CD$ at $R$ . Prove $\angle XPY = \angle XQY + \angle XRY$ .

What is the largest amount of elements that can be taken from the set $\{1, 2, ... , 2012, 2013\}$ , such that within them there are no distinct three, say $a$ , $b$ ,and $c$ , such that $a$ is a divisor or multiple of $b-c$ ?

A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$ , $1 \times n \times 1$ , and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$ , $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.

Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$ , define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$ , where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$ . Show some number $i$ , amongst 1, 2, 3, and 4 satisfies $$\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}$$