VTMO 2016 — Grade 8 — final

Find all real $\left(x , y\right)$ satisfying $\left(4 x^{2} + 6 x + 4\right) \left(4 y^{2} - 12 y + 25\right) = 28$.

Given $5 x + 4 y + 3 z + 6 t = 20$ and $x \ge y \ge z \ge t \ge 0$, find the minimum and maximum 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.

The words $0$; $01$; $0110$; $01101001$; … are formed by appending the bitwise complement. The $12$ th word has $2048$ digits. What is its $2016$ th digit?

Rectangle $ABCD$ has $AB = 5 , BC = 2$. $M$ lies on $AB$ with $AM = 1 , N$ on $DC$ with $DN = 2$. $E$ is $D$ reflected in $M , F =$ EN $\cap CM$. $a$)Prove $CM \perp DM$. $b$)For $K = DF \cap CE$, find $DK$.

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.