AT-OEMO-REG 2014

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases}$

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$ . (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$ , let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$ .Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$ , respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$ , respectively.In a given triangle $ABC$ , determine all points $P$ for which the triangles $PRD$ , $PEQ$ and $PTE$ have the same area.