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 \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

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 $n$

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.

Let $x,y,z$ be distinct positive integers such that $(y+z)(z+x)=(x+y)^2$ . Show that \[x^2+y^2>8(x+y)+2(xy+1).\] (Paolo Leonetti)

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.

For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.

Two distinct real numbers are written on each vertex of a convex $2012-$gon. Show that we can remove a number from each vertex such that the remaining numbers on any two adjacent vertices are different.

1 and only round Hi everyone, In a week there will be the 4th edition of Oliforum Contest: it's an individual and telematic contest, with some training problems at national/imo level. There will be no prize, but it can be useful to some forumers around here. All the details in the following list: 1- The contest is made by a unique round, starting in the morning of 1 october, and ending at 23:59 of 6 october (Rome meridian +1GTM): that is, 6 days. 2- Every problem will have some points, from 0 to 7: in particular, 6 for all correct solutions and 7 only for clear and original ones. 3- Everyone can send his solutions: there are no age contraints. 4- Non-olympic arguments cannot be used, otherwise the solution is not going to be valid. 5- It's enough to send the solution by mail, that you can find below. Enrollment is not necessary. 6- Time of incoming solutions will be considered only for ex-aequo positions. 7- How to send solutions. - You need to send solutions to the following mail: $\text{ leonetti.paolo (AT) gmail.com }$. - Solutions need to be attached in a unique attached .pdf file with a reasonable size. - The .pdf file must be written with $\LaTeX$ or in a way that can be easily understood. - You have to rename the .pdf file with the nickname that you have here on MathLinks. - Do not write anything else in the mail, unless strictly necessary. - Try to be clear everytime. Previous editions: { Ed. 2008, round 1, round 2, round 3 } , { Ed. 2009, round 1, round 2 }, { Ed. 2012 round 1 }. I'll put below the list of problems on the morning of 1 october 2013: later, the list with all people who sent correctly their solutions will be added. Any comment and/or suggestion will be welcome. Thanks for the attention Paolo Leonetti Italian version