VTMO 2016 — Grade 9 — final

Let positive $a , b , c , d$ satisfy $abcd = 1$. Prove $a ) \left(ac + bd\right) \left(ad + bc\right) \ge \left(a + b\right) ( c + d$); $b ) \frac{1}{a \left(1 + b\right)} + \frac{1}{b \left(1 + c\right)} + \frac{1}{c \left(1 + d\right)} + \frac{1}{d \left(1 + a\right)} \ge 2$.

$a$)Find integer solutions to $\left(3 x^{2} + y^{2} - 4 y - 17\right)^{3} - \left(2 x^{2} + 2 y^{2} - 4 y - 6\right)^{3} = \left(x^{2} - y^{2} - 11\right)^{3}$. $b$)If $5 x + 4 y + 3 z + 6 t = 20$ and $x \ge y \ge z \ge t \ge 0$, find the extrema of $x + y + z + t$.

Positive integers $a , b , c , d$ satisfy $a \gt b , c \gt d$ and $a + b + c + d = ab - cd$. Prove $a + c$ is composite.

Initially $n$ coins all show heads. An operation selects three consecutive coins, leaves the middle one, and flips the outer two. Find all $n$ permitting all tails.

$ABCD$ is a square centered at $O$. $M$ lies inside $AB$; the line through $M$ parallel to $BD$ meets $AC$ at $N$. Let $E = DN \cap CM$ and $F = DN \cap AB$. $a$)Show $\angle DEC$ is constant. $b$)Show $E$ lies on circle $\left(O , OA\right)$. $c$)Maximize $MF$.

In quadrilateral $ABCD , M$ on $AB , N$ on $CD$ satisfy $\frac{BM}{BA} = \frac{DN}{DC} = \frac{1}{3}$. Construct parallelogram $ABCE$. Let $I , J , O$ be midpoints of $AC , BD , MN$. Prove $a ) DE = 2 IJ$; $b ) O , I , J$ are collinear.