VTMO 2015 — Grade 8 — final

$f \left(x\right) = x^{2016} + 2 x^{2015} - 3 x + 4 , g \left(x\right) = x^{2} + 3 x + 2$. Find the remainder of $f$ divided by $g$.

For pairwise distinct $a , b , c$, simplify $\frac{\frac{1}{\left(a - b\right)^{2}} + \frac{1}{\left(b - c\right)^{2}} + \frac{1}{\left(c - a\right)^{2}}}{\left[\frac{1}{a - b} + \frac{1}{b - c} + \frac{1}{c - a}\right]^{2}}$.

$x = \frac{b^{2} + c^{2} - a^{2}}{2 bc} , y = \frac{a^{2} - \left(b - c\right)^{2}}{\left(b + c\right)^{2} - a^{2}}$. Find $P = x + y + xy$.

Acute $ABC$ has $AB \lt AC$, angle bisector $AD$ and median $AM , D , M$ on $BC$. $CH \perp AD$ at $H$; $HM$ meets $AC$ at $N , CH$ meets $AM$ at $E$, and $ED$ meets $AB$ at $F$. Prove $a ) N$ is midpoint $AC$; $b ) DFA$ is isosceles.

Tanker $B$ starts $100 \,\mathrm{km}$ north of cargo ship $A$. A sails northwest at $20 \frac{\,\mathrm{km}}{h}$ and $B$ southeast at $30 \frac{\,\mathrm{km}}{h}$. How far has A traveled when they are closest?

Four balls have masses $1 g , 2 g , 3 g , 5 g$. An electronic scale weighs any combined mass. Show that exactly two weighings can identify every ball.