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[Solutions] European Mathematical Cup 2014

Junior Division

  1. Which of the following claims are true, and which of them are false? If a fact is true you should prove it, if it isn't, find a counterexample.
    a) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $.
    b) Let $a,b,c$ be real numbers such that $ a^{2014} + b^{2014} + c^{2014} = 0 $. Then $ a^{2015} + b^{2015} + c^{2015} = 0 $.
    c) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $ and $ a^{2015} + b^{2015} + c^{2015} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $.
  2. In each vertex of a regular $n$-gon $A_1A_2...A_n$ there is a unique pawn. In each step it is allowed to move all pawns one step in the clockwise direction or to swap the pawns at vertices $A_1$ and $A_2$. Prove that by a finite series of such steps it is possible to swap the pawns at vertices
    a) $A_i$ and $A_{i+1}$ for any $ 1 \leq i < n$ while leaving all other pawns in their initial place
    b) $A_i$ and $A_j$ for any $ 1 \leq i < j \leq n$ leaving all other pawns in their initial place.
  3. Let $ABC$ be a triangle. The external and internal angle bisectors of $\angle CAB$ intersect side $BC$ at $D$ and $E$, respectively. Let $F$ be a point on the segment $BC$. The circumcircle of triangle $ADF$ intersects $AB$ and $AC$ at $I$ and $J$, respectively. Let $N$ be the mid-point of $IJ$ and $H$ the foot of $E$ on $DN$. Prove that $E$ is the incenter of triangle $AHF$, or the center of the excircle.
  4. Find all functions $f$ from positive integers to themselves such that: $f(mn)=f(m)f(n)$ for all positive integers $m$, $n$; $\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

Senior Division

  1. Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$ Note. For positive integer $d(a)$ denotes number of positive divisors of $a$
  2. Jeck and Lisa are playing a game on table dimensions $m \times n$ , where $m,n>2$. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules: Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table. Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table. Player which cannot put his figurine loses. For which pairs of $(m,n)$ Lisa has winning strategy?
  3. Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram
  4. Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $$f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y)),\,\forall x,y\in \mathbb{R}$$

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MOlympiad: [Solutions] European Mathematical Cup 2014
[Solutions] European Mathematical Cup 2014
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