BxMO 2019

a) Let $a, b, c, d$ be real numbers with $0 \leqslant a, b, c, d \leqslant 1$. Prove that

$$a b(a-b)+b c(b-c)+c d(c-d)+d a(d-a) \leqslant \frac{8}{27}$$

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.

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?

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 $\angle B A C$ intersects $\Gamma_{1}$ again at $X$ and $\Gamma_{2}$ again at $Y$. Prove that the interior angle bisector of $\angle B Z C$ passes through the circumcentre of $\triangle X Y Z$.

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 $\angle P Q S$ is the line that divides $\angle P Q S$ into two equal angles, while the exterior angle bisector of $\angle P Q S$ is the line that divides $\angle R Q S$ into two equal angles.

An integer $m>1$ is rich if for any positive integer $n$, there exist positive integers $x, y, z$ such that $n=m x^{2}-y^{2}-z^{2}$. An integer $m>1$ is poor if it is not rich. a) Find a poor integer. b) Find a rich integer. a) Solution I. We will show that $m=4$ is poor. If $y$ and $z$ are both even, we have $4 x^{2}-y^{2}-z^{2} \equiv$ $0-0-0=0(\bmod 4)$. If $y$ is even and $z$ is odd or the other way around, then $4 x^{2}-y^{2}-z^{2} \equiv$ $0-0-1 \equiv 3(\bmod 4)$. If $y$ and $z$ are both odd, we have $4 x^{2}-y^{2}-z^{2} \equiv 0-1-1 \equiv 2(\bmod 4)$. Hence it is impossible to write any integer $n \equiv 1(\bmod 4)$ as $n=m x^{2}-y^{2}-z^{2}$. So $m=4$ is poor.