For all IMO 2016 aspirants

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

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

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


(2)(2) Show that if pp is a prime and 0m<n<p0 \le m < n < p, then

 (np+mmp+n)(1)m+n+1p(modp2)\huge\ \left( \begin{matrix} np+m \\ mp+n \end{matrix} \right) \equiv { \left( -1 \right) }^{ m+n+1 }p \pmod{{ p }^{ 2 }}.


(3)(3) Given two circles that intersect at XX and YY, prove that there exist four points with the following property. For any circle \wp tangent to the two given circles, we let AA and BB be the points of the tangency an CC and DD the intersections of \wp with the line XYXY. Then each of the lines AC,AD,BC,BDAC, AD, BC, BD passes through one of these four points.


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

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

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


(5)(5) Let aa, bb, cc be positive real numbers. Prove that

 (ab+c)2016(a2016b2016+b2015c)(ac+a)2015\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)(6) Let II be the in-center of triangle ABCABC. It is known that for every point M(AB)M \in (AB), one can find the points N(BC)N \in (BC) and P(AC)P \in (AC) such that II is the centroid of the triangle MNPMNP. Prove that ABCABC is an equilateral triangle.

Note by Priyanshu Mishra
3 years, 10 months ago

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I dunno hows this

Jun Arro Estrella - 3 years, 10 months ago

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Please explain clearly what you want to say.

Priyanshu Mishra - 3 years, 10 months ago

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any hints to the last question?

Neel Khare - 2 years, 11 months ago

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