Olympiad problems

Oliforum Contest IV 2013, 6

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.

Oliforum Contest IV 2013, 3

Tags: combinatorics proposed , combinatorics , Contest 8

Given an integer $n$ greater than $1$, suppose $x_1,x_2,\ldots,x_n$ are integers such that none of them is divisible by $n$, and neither their sum. Prove that there exists atleast $n-1$ non-empty subsets $\mathcal I\subseteq \{1,\ldots, n\}$ such that $\sum_{i\in\mathcal I}x_i$ is divisible by $n$

Oliforum Contest IV 2013, 1

Tags: modular arithmetic , number theory , greatest common divisor , number theory proposed , Contest 8

Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

Oliforum Contest IV 2013, 4

Tags: algebra , polynomial , quadratics , number theory proposed , number theory , Contest 8

Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.