OSN 2014

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?

For some positive integers $m,n$ , the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$ . Determine all possible values of $m-n$ .

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$ . Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$ . Construct the parallelograms $AEDK$ and $BECL$ . Prove that $EF$ passes through the midpoint of the segment $KL$ .

Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$ . Prove that if $p$ divides $q$ then $a_p \le a_q$ .

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$ . Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$ , and $N$ is on $AC$ such that $\angle NDA = \angle ACB$ . If $AD$ and $MN$ intersect on $P$ , prove that $AD^3 = AB \cdot AC \cdot AP$ .

Suppose that $k,m,n$ are positive integers with $k \le n$ . Prove that: $$\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1$$

A positive integer is called beautiful if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$ . A positive integer that is not beautiful is ugly.

a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.