VN TST 2016

Find all positive integers $a,n$, with $a\gt 2$, such that every prime divisor of $a^n-1$ also divides $a^{3^{2016}}-1$.

Let $A$ be a set of $2000$ distinct integers and $B$ a set of $2016$ distinct integers. Let $K$ count the ordered pairs $(m,n)\in A\times B$ satisfying $|m-n|\le 1000$. Find the largest possible value of $K$.

Let triangle \(ABC\) be inscribed in a fixed circle \((O)\), with \(B,C\) fixed and \(A\) moving on one fixed arc \(BC\). The internal angle bisectors \(AD,BE,CF\) meet at \(I\). The circle through \(D\), tangent to line \(OA\) at \(A\), meets \((O)\) again at \(G\). Lines \(GE,GF\) meet \((O)\) again at \(M,N\), respectively, and \(H=BM\cap CN\).
(a) Prove that \(AH\) passes through a fixed point.
(b) Let \(K,L\) be the second intersections of \(BE,CF\) with \((O)\), and let \(P=AH\cap KL\). Choose \(Q\in EF\) with \(QP=QI\). Let \(J\ne I\) be the second intersection of the line through \(I\) perpendicular to \(IQ\) with \((BIC)\). Prove that the midpoint of \(IJ\) lies on a fixed circle.
As is necessary for the construction, consider positions in which all named circles, lines, and intersections exist and are distinct as used.

Let \(ABC\) be an acute triangle satisfying
\[\angle ACB\lt \angle ABC\lt \angle ACB+\frac12\angle BAC.\]
Choose \(D\) on segment \(BC\) so that \(\angle ADC=\angle ACB+\tfrac12\angle BAC\). The tangent at \(A\) to the circumcircle of \(ABC\) meets line \(BC\) at \(E\). The internal bisector of \(\angle AEB\) meets \(AD\) at \(G\) and the circle \((ADE)\) again at \(F\ne E\). Put \(H=DF\cap AE\).
(a) Prove that the circles with diameters \(AE,DF,GH\) have a common point.
(b) Choose \(K\) on the external angle-bisector line of \(\angle BAC\) and \(M\) on ray \(AC\) such that \(KB=KD=KM\). Similarly choose \(L\) on that external bisector and \(N\) on ray \(AB\) such that \(LC=LD=LN\). Let \(I\) be the midpoint of \(BC\), and let the circle \((MNI)\) meet \(BC\) again at \(P\ne I\). Prove that \(BM,CN,AP\) are concurrent.

Let $n\ge3$ and $a_1,\ldots,a_n\in\{0,1\}$. Let $S_i=(a_i,a_{i+1},\ldots,a_n,a_1,\ldots,a_{i-1})$ be the $n$ cyclic shifts. For a binary tuple $r=(b_1,\ldots,b_n)$ define $\omega(r)=\sum_{j=1}^n b_j2^{n-j}$. Suppose $\omega(S_1),\ldots,\omega(S_n)$ take exactly $k$ distinct values.
1. Prove $k\mid n$ and $\frac{2^n-1}{2^k-1}\mid\omega(S_i)$ for every $i$.
2. If $M$ and $m$ are their maximum and minimum, prove
$$M-m\ge\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}.$$

Let $a_1,\ldots,a_{16}$ be distinct real numbers. For each real polynomial $P$, put $V(P)=\sum_{i=1}^{16}P(a_i)$. Prove there exists exactly one monic polynomial $Q$ of degree $8$ satisfying both:
1. $V(QP)=0$ for every polynomial $P$ of degree less than $8$;
2. $Q$ has $8$ real roots, counted with multiplicity.