NMC 2015

Let $A B C$ be a triangle and $\Gamma$ the circle with diameter $A B$. The bisectors of $\angle B A C$ and $\angle A B C$ intersect $\Gamma$ (also) at $D$ and $E$, respectively. The incircle of $A B C$ meets $B C$ and $A C$ at $F$ and $G$, respectively. Prove that $D, E, F$ and $G$ are collinear.

Find the primes $p, q, r$, given that one of the numbers $p q r$ and $p+q+r$ is 101 times the other.

Let $n>1$ and $p(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$ be a polynomial with $n$ real roots (counted with multiplicity). Let the polynomial $q$ be defined by

$$q(x)=\prod_{j=1}^{2015} p(x+j)$$

We know that $p(2015)=2015$. Prove that $q$ has at least 1970 different roots $r_{1}, \ldots, r_{1970}$ such that $\left|r_{j}\right|<2015$ for all $j=1, \ldots, 1970$.

An encyclopedia consists of 2000 numbered volumes. The volumes are stacked in order with number 1 on top and 2000 in the bottom. One may perform two operations with the stack:

(i) For $n$ even, one may take the top $n$ volumes and put them in the bottom of the stack without changing the order.

(ii) For $n$ odd, one may take the top $n$ volumes, turn the order around and put them on top of the stack again.

How many different permutations of the volumes can be obtained by using these two operations repeatedly?