PMO 2016

The operations below can be applied on any expression of the form $ax^2+bx+c$. $(\text{I})$ If $c \neq 0$, replace $a$ by $4a-\frac{3}{c}$ and $c$ by $\frac{c}{4}$. $(\text{II})$ If $a \neq 0$, replace $a$ by $-\frac{a}{2}$ and $c$ by $-2c+\frac{3}{a}$. $(\text{III}_t)$ Replace $x$ by $x-t$, where $t$ is an integer. (Different values of $t$ can be used.)

Is it possible to transform $x^2-x-6$ into each of the following by applying some sequence of the above operations? $(\text{a})$ $5x^2+5x-1$ $(\text{b})$ $x^2+6x+2$

Prove that the arithmetic sequence $5, 11, 17, 23, 29, \ldots$ contains infinitely many primes.

Let $n$ be any positive integer. Prove that $$\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}$$.

Two players, $A$ (first player) and $B$, take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove $s$ chips from the remaining pile of chips, where $s \in \{ 2,4,5 \}$. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?

Pentagon $ABCDE$ is inscribed in a circle. Its diagonals $AC$ and $BD$ intersect at $F$. The bisectors of $\angle BAC$ and $\angle CDB$ intersect at $G$. Let $AG$ intersect $BD$ at $H$, let $DG$ intersect $AC$ at $I$, and let $EG$ intersect $AD$ at $J$. If $FHGI$ is cyclic and $$JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,$$ prove that $G$, $F$ and $E$ are collinear.