$n$ is a positive integer. The number of solutions of $x^2+2016y^2=2017^n$ is $k$. Write $k$ with $n$.

A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$. Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$. Let $T$ be the area of $\triangle DEF$. Prove the following inequality. $$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$

For a positive integer $n$, $S_n$ is the set of positive integer $n$-tuples $(a_1,a_2, \cdots ,a_n)$ which satisfies the following. (i). $a_1=1$. (ii). $a_{i+1} \le a_i+1$. For $k \le n$, define $N_k$ as the number of $n$-tuples $(a_1, a_2, \cdots a_n) \in S_n$ such that $a_k=1, a_{k+1}=2$. Find the sum $N_1 + N_2+ \cdots N_{k-1}$.

A non-isosceles triangle $\triangle ABC$ has incenter $I$ and the incircle hits $BC, CA, AB$ at $D, E, F$. Let $EF$ hit the circumcircle of $CEI$ at $P \not= E$. Prove that $\triangle ABC = 2 \triangle ABP$.

For a positive integer $n$, there are $n$ positive reals $a_1 \ge a_2 \ge a_3 \cdots \ge a_n$. For all positive reals $b_1, b_2, \cdots b_n$, prove the following inequality. $$\frac{a_1b_1+a_2b_2 + \cdots +a_nb_n}{a_1+a_2+ \cdots a_n} \le \text{max}\{ \frac{b_1}{1}, \frac{b_1+b_2}{2}, \cdots, \frac{b_1+b_2+ \cdots +b_n}{n} \}$$

Let $N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}$. Prove that there are $(b_1+1)(b_2+1)\ldots(b_k+1)$ number of $n$s which satisfies these two conditions. $\frac{n(n+1)}{2}\le N$, $N-\frac{n(n+1)}{2}$ is divided by $n$.

A subset $S \in \{0, 1, 2, \cdots , 2000\}$ satisfies $|S|=401$. Prove that there exists a positive integer $n$ such that there are at least $70$ positive integers $x$ such that $x, x+n \in S$