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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, 9 months ago

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I dunno hows this

Jun Arro Estrella - 1 year, 9 months ago

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Please explain clearly what you want to say.

Priyanshu Mishra - 1 year, 9 months ago

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any hints to the last question?

Neel Khare - 10 months, 3 weeks ago

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