CWMO 2017

Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$ . Show that $p<2n$ .

Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$ Find the greatest possible value of $n$ .

D is the a point on BC，I1 is the heart of a triangle ABD, I2 is the heart of a triangle ACD，O1 is the Circumcenter of triangle AI1D, O2 is the Circumcenter of the triangle AI2D，P is the intersection point of O1I2 and O2I1，Prove: PD is perpendicular to BC.

Attachments:

Let $n$ and $k$ be given integers such that $n\ge k\ge 2$ . Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?

Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$ , there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$ . Find the number of $9$ -tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.

In acute triangle $ABC$ , let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$ . Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$ .

Let $n=2^{\alpha} \cdot q$ be a positive integer, where $\alpha$ is a nonnegative integer and $q$ is an odd number. Show that for any positive integer $m$ , the number of integer solutions to the equation $x_1^2+x_2^2+\cdots +x_n^2=m$ is divisible by $2^{\alpha +1}$ .

Let $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$ . Prove that $$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min \{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$