[Solutions] Benelux Mathematical Olympiad 2019

  1. a) Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that $$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$ b) Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality.
  2. Pawns and rooks are placed on a $2019\times 2019$ chessboard, with at most one piece on each of the $2019^2$ squares. A rook can see another rook if they are in the same row or column and all squares between them are empty. What is the maximal number $p$ for which $p$ pawns and $p+2019$ rooks can be placed on the chessboard in such a way that no two rooks can see each other?
  3. Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A\neq Z$). Let $B$ be the centre of $\Gamma_1$ and let $C$ be the centre of $\Gamma_2$. The exterior angle bisector of $\angle{BAC}$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Prove that the interior angle bisector of $\angle{BZC}$ passes through the circumcenter of $\triangle{XYZ}$. (For points $P$, $Q$, $R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle{PQS}$ is the line that divides $\angle{PQS}$ into two equal angles, while the exterior angle bisector of $\angle{PQS}$ is the line that divides $\angle{RQS}$ into two equal angles.)
  4. An integer $m>1$ is rich if for any positive integer $n$, there exist positive integers $x,y,z$ such that $n=mx^2-y^2-z^2$. An integer $m>1$ is poor if it is not rich.
    a) Find a poor integer.
    b) Find a rich integer.
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