Prove for all positive reals a,b,c,d: $ \frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b} \geq 0$

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove $ AB+BC+CA\geq 8DE$

Prove that there are infinite many positive integers $ n$ such that $ n^2+1\mid n!$, and infinite many of those for which $ n^2+1 \nmid n!$.

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

In each field of 2009*2009 table you can write either 1 or -1. Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column. Is it possible to write the numbers in such a way that $ \sum_{i=1}^{2009}{Ai}+ \sum_{i=1}^{2009}{Bi}=0$?

It is given a convex quadrilateral $ ABCD$ in which $ \angle B+\angle C < 180^0$. Lines $ AB$ and $ CD$ intersect in point E. Prove that $ CD*CE=AC^2+AB*AE \leftrightarrow \angle B= \angle D$

Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a-b^c\mid=1$

Solve in the set of real numbers: \[ 3\left(x^2 + y^2 + z^2\right) = 1, \] \[ x^2y^2 + y^2z^2 + z^2x^2 = xyz\left(x + y + z\right)^3. \]

Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad = bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.

A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.

Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n + 1$, $ n + 2$, $ n + 3$.