VTMO 2021 — Grade 8 — cup

Find primes $p$ admitting integers $a , b$ with $\frac{1}{p} = \frac{1}{a^{2}} + \frac{1}{b^{2}}$.

Find real $a , b$ satisfying $| ax$+by $| + | bx + ay | = | x | + | y |$ for all $x , y$.

For positive $n , n^{3} + 3 n^{2} + 3 n$ ends $4$. Find the final digit of $31 n^{2} + n + 2021$.

$A = 15 \times 7$ rectangle is tiled with $L$ triominoes and $2 \times 2$ squares as shown. The statement lacks a question after the figures; the supplied solution studies a square-tile bound.

$a$)In convex $ABCD , I$ midpoint $AC , M$ on $AC , AM = \sqrt{2}$ AI. Parallels from $M$ to $BC , DC$ meet $AB , AD$ at $E , F$. Prove $S_{IEF} = \frac{S_{MDB}}{2}$. $b$)Acute $ABC$ has median $AM$, altitude $BE , K = AM \cap BE , AM = BE$. Prove $AC = \sqrt{3} BK$.