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# 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.

Note by Priyanshu Mishra
1 year, 7 months ago

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I dunno hows this · 1 year, 7 months ago

Please explain clearly what you want to say. · 1 year, 7 months ago