Olympiad problems

2002 Olympic Revenge, 4

Find all pairs $(m,n)$ of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having $m$ sides, and the other having $n$ sides.

2002 Olympic Revenge, 7

Tags: olympic revenge 2002 , number theory unsolved , factorial , number theory

Show that \[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\] is an integer, for any positive integer $n$.

2002 Olympic Revenge, 3

Tags: inequalities , four variables , euclidean distance , olympic revenge 2002 , olympic revenge

Show that if $x,y,z,w$ are positive reals, then \[ \frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w) \]

2002 Olympic Revenge, 1

Tags: function , olympic revenge , algebra , olympic revenge 2002

Show that there is no function $f:\mathbb{N}^* \rightarrow \mathbb{N}^*$ such that $f^n(n)=n+1$ for all $n$ (when $f^n$ is the $n$th iteration of $f$)

2002 Olympic Revenge, 2

Tags: olympic revenge 2002 , geometry proposed , circles , geometry , circumcircle

$ABCD$ is a inscribed quadrilateral. $P$ is the intersection point of its diagonals. $O$ is its circumcenter. $\Gamma$ is the circumcircle of $ABO$. $\Delta$ is the circumcircle of $CDO$. $M$ is the midpoint of arc $AB$ on $\Gamma$ who doesn't contain $O$. $N$ is the midpoint of arc $CD$ on $\Delta$ who doesn't contain $O$. Show that $M,N,P$ are collinear.

2002 Olympic Revenge, 5

Tags: olympic revenge 2002 , olympic revenge , combinatorics

In a "Hanger Party", the guests are initially dressed. In certain moments, the host chooses a guest, and the chosen guest and all his friends will wear its respective clothes if they are naked, and undress it if they are dressed. It is possible that, in some moment, the guests are naked, independent of their mutual friendships? (Suppose friendship is reciprocal.)