VTMO 2021 — Grade 9 — cup

Find integers $k$ such that both roots of $x^{2} - 2$ kx $+ 3 k + 2 = 0$ are integers.

Find real $\left(a , b , c\right)$ such that $| ax$+by $+ cz | + | ay + bz + cx | + | az + bx + cy | = | x | + | y | + | z |$ for all $x , y , z$.

For pairwise distinct real $a , b , c$, minimize $P = \frac{\left(a - b\right)^{4}}{\left(b - c\right)^{2} \left(c - a\right)^{2}} + \frac{\left(b - c\right)^{4}}{\left(c - a\right)^{2} \left(a - b\right)^{2}} + \frac{\left(c - a\right)^{4}}{\left(a - b\right)^{2} \left(b - c\right)^{2}}$.

Start with piles of $51 , 49 , 5$ stones. Each move merges two piles or halves an even pile. What is the smallest obtainable pile?

Square $ABCD$ is inscribed in $\left(O\right)$. $M$ is the midpoint of minor arc $CD$, and $N \in AB$ with $NA = 2 NB$. $MN$ meets $\left(O\right)$ again at $E$; $F$ is $E$ reflected across $AB$. Prove $FA = 2 FB$ and $DF = 3 FB$.