OLYMPIC-REVENGE 2017

Prove that does not exist positive integers $a$ , $b$ and $k$ such that $4abk-a-b$ is a perfect square.

Let $\triangle$ $ABC$ a triangle with circumcircle $\Gamma$ . Suppose there exist points $R$ and $S$ on sides $AB$ and $AC$ , respectively, such that $BR=RS=SC$ . A tangent line through $A$ to $\Gamma$ meet the line $RS$ at $P$ . Let $I$ the incenter of triangle $\triangle$ $ARS$ . Prove that $PA=PI$

Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$ $:$ { $1,2,...n$ } $\rightarrow$ { $1,2,...,n$ } and a binary function $C:$ { $1,2,...,n$ } $\rightarrow$ { $0,1$ } "revengeful" if it satisfies the two following conditions: $1)$ For every $i$ $\in$ { $1,2,...,n$ }, there exist $j$ $\in$ $S_{i}=$ { $i, \pi(i),\pi(\pi(i)),...$ } such that $C(j)=1$ . $2)$ If $C(k)=1$ , then $k$ is one of the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$ , where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$ , for every positive integer $t$ .

Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$ . Determine $\frac{V}{P}$ .

Let $f:\mathbb{R}_{+}^{*}$ $\rightarrow$ $\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$ . Prove that: $f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$ , for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$ .