Greece Mathematical Olympiad 2014

Find all the polynomials with real coefficients which satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$$ for all $x\in \mathbb{R}$. Find all the int...

  1. Find all the polynomials with real coefficients which satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$$ for all $x\in \mathbb{R}$.
  2. Find all the integers $n$ for which $\dfrac{8n-25}{n+5}$ is cube of a rational number.
  3. For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\dfrac{S_1}{S_2}=\dfrac{39}{64}.$
  4. We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.



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MATHEMATICAL OLYMPIAD PROBLEMS: Greece Mathematical Olympiad 2014
Greece Mathematical Olympiad 2014
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