XK-TST 2017

Find all positive integers $(a, b)$ , such that $\frac{a^2}{2ab^2-b^3+1}$ is also a positive integer.

Prove that there doesn't exist any function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that : $f(f(n-1)=f(n+1)-f(n)$ , for every natural $n\geq2$

If $a$ and $b$ are positive real numbers with sum $3$ , and $x, y, z$ positive real numbers with product $1$ , prove that : $(ax+b)(ay+b)(az+b)\geq 27$

For every $n \in \mathbb{N}_{0}$ , prove that $\sum_{k=0}^{\left[\frac{n}{2} \right]}{2}^{n-2k} \binom{n}{2k}=\frac{3^{n}+1}{2}$

Given triangle $ABC$ . Let $P$ , $Q$ , $R$ , be the tangency points of inscribed circle of $\triangle ABC$ on sides $AB$ , $BC$ , $AC$ respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as $P',Q'$ and $R'$ . Prove that $AP'$ , $BQ'$ , and $CR'$ are concurrent.