VTMO 2013 — Grade 8 — final

$a + b + c = 3 , a^{2} + b^{2} + c^{2} = 7 , a^{3} + b^{3} + c^{3} = 15$. Find a $^{4} + b^{4} + c^{4}$.

For $a \ge b \ge c \gt 0$, prove $\left(a - b + c\right) \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \ge 1$.

Find all integer pairs $\left(x , y\right)$ satisfying $2 x^{2} - xy + 3 x - 2 y = 5$.

$ABC$ is right at $A , AC = 2 AB$, and $M$ midpoint $AB$. The perpendicular to $CM$ through A meets $BC$ at $D$. Find $\frac{BD}{DC}$.

$ABC$ has $BAC = 120 ^{\circ}$, with internal angle bisectors $AD , BE , CF$ and $D$ on $BC , E$ on $AC , F$ on $AB$. Prove $DE \perp DF$.