For all IMO 2016 aspirants

Hello INMO, USAMO and all national olympiads awardees, try these problems and post solutions , specially for IMO 2016:

$(1)$ Find all functions $f, g : R \rightarrow R$ which satisfy the equation:

$(x - y)f(z) + (y - z)f(x) + (z - x)f(y) = g(x + y + z)$, for all real numbers $x, y, z$ such that $x \neq y, y \neq z, z \neq x$.

$(2)$ Show that if $p$ is a prime and $0 \le m < n < p$, then

$\huge\ \left( \begin{matrix} np+m \\ mp+n \end{matrix} \right) \equiv { \left( -1 \right) }^{ m+n+1 }p \pmod{{ p }^{ 2 }}$.

$(3)$ Given two circles that intersect at $X$ and $Y$, prove that there exist four points with the following property. For any circle $\wp$ tangent to the two given circles, we let $A$ and $B$ be the points of the tangency an $C$ and $D$ the intersections of $\wp$ with the line $XY$. Then each of the lines $AC, AD, BC, BD$ passes through one of these four points.

$(4)$ Find all integral solutions to following equations:

$(a)$ $\huge\ { y }^{ { y }^{ { y }^{ y } } } + { x }^{ { x }^{ x } } = { \left( x!.y! \right) }^{ 2016 }$,

$(b)$ $\huge\ { a }^{ x } + { b }^{ { y }^{ z } } + { c }^{ { z }^{ { x }^{ d } } } + { d }^{ { a }^{ { y }^{ { z }^{ x } } } } = { \left( abcd \right) }^{ xyz }$.

$(5)$ Let $a$, $b$, $c$ be positive real numbers. Prove that

$\large\ { \left( \sum { \frac { a }{ b+c } } \right) }^{ 2016 } \le \left( \sum { \frac { { a }^{ 2016 } }{ { b }^{ 2016 } + { b }^{ 2015 }c } } \right) { \left( \sum { \frac { a }{ c + a } } \right) }^{ 2015 }$.

$(6)$ Let $I$ be the in-center of triangle $ABC$. It is known that for every point $M \in (AB)$, one can find the points $N \in (BC)$ and $P \in (AC)$ such that $I$ is the centroid of the triangle $MNP$. Prove that $ABC$ is an equilateral triangle.