VN-TST 2016

Find all $a,n\in\mathbb{Z}^+$ ( $a>2$ ) such that each prime divisor of $a^n-1$ is also prime divisor of $a^{3^{2016}}-1$

Let $A$ be a set contains $2000$ distinct integers and $B$ be a set contains $2016$ distinct integers. $K$ is the numbers of pairs $(m,n)$ satisfying $$\begin{cases} m\in A, n\in B |m-n|\leq 1000 \end{cases}$$ Find the maximum value of $K$

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$ , $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$ . The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ( $G \ne A$ ). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$ . Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$ . On $EF$ take $Q$ for $QP = QI$ . Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$ . Prove that the midpoint of $IJ$ belongs to a fixed circle.

Attachments:

Given an acute triangle $ABC$ satisfying $\angle ACB<\angle ABC<\angle ACB+\dfrac{\angle BAC}{2}$ . Let $D$ be a point on $BC$ such that $\angle ADC=\angle ACB+\dfrac{\angle BAC}{2}$ . Tangent of circumcircle of $ABC$ at $A$ hits $BC$ at $E$ . Bisector of $\angle AEB$ intersects $AD$ and $(ADE)$ at $G$ and $F$ respectively, $DF$ hits $AE$ at $H.$ a) Prove that circle with diameter $AE,DF,GH$ go through one common point. b) On the exterior bisector of $\angle BAC$ and ray $AC$ given point $K$ and $M$ respectively satisfying $KB=KD=KM$ , On the exterior bisector of $\angle BAC$ and ray $AB$ given point $L$ and $N$ respectively satisfying $LC=LD=LN.$ Circle throughs $M,N$ and midpoint $I$ of $BC$ hits $BC$ at $P$ ( $P\neq I$ ). Prove that $BM,CN,AP$ concurrent.

Given $n$ numbers $a_1,a_2,...,a_n$ ( $n\geq 3$ ) where $a_i\in\{0,1\}$ for all $i=1,2.,,,.n$ . Consider $n$ following $n$ -tuples $$\begin{aligned} S_1 & =(a_1,a_2,...,a_{n-1},a_n) S_2 & =(a_2,a_3,...,a_n,a_1) & \vdots S_n & =(a_n,a_1,...,a_{n-2},a_{n-1}).\end{aligned}$$ For each tuple $r=(b_1,b_2,...,b_n)$ , let $$\omega (r)=b_1\cdot 2^{n-1}+b_2\cdot 2^{n-2}+\cdots+b_n.$$ Assume that the numbers $\omega (S_1),\omega (S_2),...,\omega (S_n)$ receive exactly $k$ different values.

a) Prove that $k|n$ and $\frac{2^n-1}{2^k-1}|\omega (S_i) \forall i=1,2,...,n.$ b) Let $$\begin{aligned} M & =\max _{i=\overline{1,n}}\omega (S_i) m & =\min _{i=\overline{1,n}}\omega (S_i). \end{aligned}$$ Prove that $$M-m\geq\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}.$$

Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$ . For each polynomial $P$ , denote $$V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}).$$ Prove that there is a monic polynomial $Q$ , $\deg Q=8$ satisfying:

i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$ .

ii) $Q$ has $8$ real roots (including multiplicity).