Problem 1
Let $O$ be the center of the circumcircle of an acute triangle $ABC$. The line $AC$ intersects the circumcircle of the triangle $ABO$ a second time at $S$. Prove that the line $OS$ is perpendicular to the line $BC$.
Problem 2
Let $ABC$ be an acute triangle with $BC > AC$. The perpendicular bisector of the segment $AB$ intersects the line $BC$ at $X$ and the line $AC$ at $Y$. Let $P$ be the projection of $X$ on $AC$ and let $Q$ be the projection of $Y$ on $BC$. Prove that the line $PQ$ intersects the segment $AB$ at its midpoint.
Problem 3
Anaelle has $2n$ stones labeled $1$, $2$, $3$, … , $2n$ as well as a red box and a blue box. She wants to put each of the $2n$ stones into one of the two boxes such that the stones $k$ and $2k$ are in different boxes for all $k = 1,2,…,n$. How many possibilities does Anaelle have to do so?
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Learn moreProblem 4
Prove that for every integer $n \geq 3$ there exist positive integers $a_1 < a_2 < … < a_n$, such that
$$ a_k | \left(a_1 + a_2 + … + a_n \right) $$
holds for every $k = 1,2,…,n$.
Problem 5
Find all positive integers $n \geq 2$, such that, for every divisor $d > 1$ of $n$, we have that $d^2 + n$ divides $n^2 + d $.
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