VTMO 2019 — Grade 8 — final

Real $a , b$ satisfy a $^{3} + b^{3} + 6 ab = 8$. What values can $a + b$ have?

$a$)Pairwise distinct nonzero $a , b , c$ satisfy $a \left(a - b\right) = b \left(b - c\right) = c \left(c - a\right)$. Find $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. $b$)Positive integers satisfy $a + b = b \left(a - c\right)$ and $c + 1 = p^{2}$ for a prime $p$. Prove $a + b$ or $ab$ is a square.

Find all integer solutions of $2^{a} + 8 b^{2} - 3^{c} = 283$.

$ABC$ is isosceles at $A , BD$ bisects $B$, and altitude $AH = \frac{BD}{2}$. Find angle $BAC$.

In convex quadrilateral $ABCD , M , N$ are midpoints of $BC , AD , P = AM \cap BN , Q = DM \cap CN$. Prove $S_{PAB} + S_{QCD} = S_{MPNQ}$.

Allow replacements $BBA$↔$A$ and $ABA$↔$B$ in words over $A , B$. Can $B A^{2019}$ be changed to $A^{2019} B$? Explain.