VTMO 2013 — Grade 9 — final

Find all real a for which $x^{2} - ax - a + 2 = 0$ has an integer root.

Let $a \ge b \ge c \gt 0$. Prove $\left(a - b + c\right) \left(\frac{1}{a} - \frac{1}{b} + \frac{1}{c}\right) \ge 1$.

Let $p \gt q \gt 2$ be consecutive primes. Prove $p + q$ has at least four positive divisors.

$ABC$ is isosceles at $A$, with altitude $AH$. Let $M$ lie inside $HC , E , F$ its projections on $AC , AB$, and $O$ the midpoint of $AM$. Prove $HO \perp EF$.

$ABCD$ circumscribes circle $\left(O\right)$, with tangency points $M , N , P , Q$ on $AB , BC , CD , DA$. Let $I = NQ \cap AC$. $a$)Prove $\frac{IA}{IC} = \frac{QA}{NC}$. $b$)Prove $AC , BD , MP , NQ$ concur.