VTMO 2026 - Grade 8 - Final - SET 4 - Self-study

In parallelogram $ABCD$, $AB>AD$. Point $E$ lies on side $AB$ with $AE=AD$. Let $F=DE\cap AC$. Take $K$ on segment $EF$, and let $M=AK\cap BC$, $N=CK\cap AB$. Prove that $AN=CM$.

Find all real pairs $(a,b)$ such that $|ax+by|+|bx+ay|=|x|+|y|$ for every pair of real numbers $x,y$.

In quadrilateral $ABCD$, let $M,N$ be the midpoints of $BC,AD$, respectively. Let $P=AM\cap BN$ and $Q=DM\cap CN$. Prove that $S_{PAB}+S_{QCD}=S_{MPNQ}$, where $S$ denotes area.

Find all prime numbers $p$ for which there exist integers $a,b$ satisfying $\frac1p=\frac1{a^2}+\frac1{b^2}$.

Find all integers $a$ for which $\frac{a^3+6}{a^2+3}$ is an integer.