VTMO 2014 — Grade 8 — final

$a + b + c = 0 , x + y + z = 0 , \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 0$. Find $A = a^{2} x + b^{2} y + c^{2} z$.

Prove that $A = \left(a + b\right) \left(a + 2 b\right) \left(a + 2015 b\right) \left(a + 2016 b\right) + 1007^{2} b^{4}$ is a square for all integers $a , b$.

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

Trapezoid $ABCD$ has $AB \parallel CD , AB \lt CD$, and extensions $DA , CB$ meeting at I. $E$ lies on $CD , IE$ meets $AB$ at $F$. $P$ lies on the ray opposite $BD$. $PF$ meets $AD$ at $M$ and $PE$ meets $BC$ at $N$. Prove $MN \parallel AB$.

$ABCD$ is a square, $E$ lies on $BC$ with $BE = \frac{BC}{4}$. The bisector of $DEB$ meets $AB$ at $F$. Prove $DF$ bisects $ADE$.

Among $N$ respondents, $9 \frac{N}{14}$ value flower color, $7 \frac{N}{12}$ value scent, and $753$ value both. Find the least $N$.