This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2

2019 Macedonia National Olympiad, 1
Tags: MMO 2019 , Macedonia , geometry , circumcircle
In an acute-angled triangle $ABC$, point $M$ is the midpoint of side $BC$ and the centers of the $M$- excircles of triangles $AMB$ and $AMC$ are $D$ and $E$, respectively. The circumcircle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. The circumcircle of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF\hspace{0.25mm} = \hspace{0.25mm} CG$ .

2019 Macedonia National Olympiad, 2
Tags: Macedonia , MMO 2019 , number theory
Let $n$ be a positive integer. If $r\hspace{0.25mm} \equiv \hspace{1mm} n\hspace{1mm} (mod\hspace{1mm} 2)$ and $r\hspace{0.10mm} \in \hspace{0.10mm} \{ 0,\hspace{0.10mm} 1 \} $, find the number of integer solutions to the system of equations $\left\{\begin{array}{l}x+y+z = r \\ \mid x \mid + \mid y \mid + \mid z \mid = n \end{array}\right.$