# [Solutions] The Mathematical Danube Competition 2015 (Senior)

1. Let $ABCD$ be a cyclic quadrangle, let the diagonals $AC$ and $BD$ cross at $O$, and let $I$ and $J$ be the incentres of the triangles $ABC$ and $ABD$, respectively. The line $IJ$ crosses the segments $OA$ and $OB$ at $M$ and $N$, respectively. Prove that the triangle $OMN$ is isosceles.
2. Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even
3. Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.
4. Given an integer $n \ge 2$ ,determine the numbers that written in the form $a_1$$a_2$$+$$a_2$$a_3$$+$$...$$a_{k-1}$$a_k$ , where $k$ is an integer greater than or equal to 2, and $a_1$ ,... $a_k$ are positive integers with sum $n$.
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