Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations \[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

$ABCDEF$ is a convex hexagon, such that $|\angle FAB| = |\angle BCD| =|\angle DEF|$ and $|AB| =|BC|,$ $|CD| = |DE|$, $|EF| = |FA|$. Prove that the lines $AD$, $BE$ and $CF$ are concurrent.

Find all primes $p$ such that the expression \[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\] is divisible by $p^3$.

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

$ABCDE$ is a regular pentagon. Determine the smallest value of the expression \[\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},\] where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$.

Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$.