Let $a,b,c$ be positive integers , such that $A=\frac{a^2+1}{bc}+\frac{b^2+1}{ca}+\frac{c^2+1}{ab}$ is, also, an integer. Proof that $\gcd( a, b, c)\leq\lfloor\sqrt[3]{a+ b+ c}\rfloor$.

Let the sequence $a_n$, $n\geq2$, $a_n=\frac{\sqrt[3]{n^3+n^2-n-1}}{n} $. Find the greatest natural number $k$ ,such that $a_2 \cdot a_3 \cdot . . .\cdot a_k <8$

The incircle of triangle $ABC$ touches $AC$ and $BC$ respectively $P$ and $Q$. Let $N$ and $M$ be the midpoints of the sides $AC$ and $BC$ respectively.$AM$ and $BP$,$BN$ and $AQ$ intersects at the points $X$ and $Y$ respectively. If the points $C,X$ and $Y$ are collinear , then prove that $CX$ is the angle bisector of $\angle ACB$.