Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.

In a acutangle triangle $ABC, \angle B>\angle C$. Let $D$ the foot of the altitude from $A$ to $BC$ and $E$ the foot of the perpendicular from $D$ to $AC$. Let $F$ a point in $DE$. Prove that $AF$ and $BF$ are perpendiculars if and only if $EF\cdot DC=BD\cdot DE$.

If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$$$T(2013)=T(2016)$$

Let $A=\{1,2,4,5,7,8,\dots\}$ the set with naturals not divisible by three. Find all values of $n$ such that exist $2n$ consecutive elements of $A$ which sum it´s $300$.

In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.

If we separate the numbers $1,2,3,4,\dots, 100$ in two lists with $$a_1<a_2<\cdots<a_{50}$$and $$b_1>b_2>\cdots>b_{50}$$ Prove that, no matter how we do the separation, $$\vert a_1-b_1\vert +\vert a_2-b_2\vert+\cdots +\vert a_{50}-b_{50}\vert=2500$$