VTMO 2017 — Grade 9 — final

A car plans an $80 \,\mathrm{km}$ trip. Rain makes the first quarter $15 \frac{\,\mathrm{km}}{h}$ slower and the rest $10 \frac{\,\mathrm{km}}{h}$ faster, keeping the planned time. Find that time.

Find primes $p , q$ satisfying $2 p + 6 q = \left(p - q\right)^{3}$.

Integers $a , b$ make $x^{2} + ax + 1 - b = 0$ have two nonzero integer roots. Prove a $^{2} + b^{2}$ is composite.

An isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB \lt CD$ is cyclic with center $O$. Let $K = AC \cap BD$; $M , N , P , Q$ are circumcenters of $KAB , KCD , KAD , KBC$. Prove $MQNP$ is a parallelogram and $KM = ON$.

$ABC$ is scalene with $BC$ its shortest side. $D \in AB , E \in AC$ satisfy $BD = CE = BC$. Let $M = BE \cap CD , MH \perp BC , K = AM \cap BC$. Prove $BH = \frac{BC + BA - AC}{2}$, and if $AB + AC = 3 BC$, prove $CK = BH$.

A number is super-even if all its digits are even. How many four-digit super-even numbers remain super-even after multiplication by $3$?