CGMO 2016

Let $n\ge 3$ be an integer. Put $n^2$ cards, each labelled $1,2,\ldots ,n^2$ respectively, in any order into $n$ empty boxes such that there are exactly $n$ cards in each box. One can perform the following operation: one first selects $2$ boxes, takes out any $2$ cards from each of the selected boxes, and then return the cards to the other selected box. Prove that, for any initial order of the $n^2$ cards in the boxes, one can perform the operation finitely many times such that the labelled numbers in each box are consecutive integers.

In $\triangle ABC, BC=a, CA=b, AB=c,$ and $\Gamma$ is its circumcircle. $(1)$ Determine a necessary and sufficient condition on $a,b$ and $c$ if there exists a unique point $P(P\neq B, P\neq C)$ on the arc $BC$ of $\Gamma$ not passing through point $A$ such that $PA=PB+PC$ . $(2)$ Let $P$ be the unique point stated in $(1)$ . If $AP$ bisects $BC$ , prove that $\angle BAC<60^{\circ}$ .

Let $m$ and $n$ are relatively prime integers and $m>1,n>1$ . Show that:There are positive integers $a,b,c$ such that $m^a=1+n^bc$ , and $n$ and $c$ are relatively prime.

Let $n$ is a positive integers , $a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ . $(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$ $(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$ for the integer $k\ge 3.$

Define a sequence $\{a_n\}$ by$$S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).$$ Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$ . For any positive integer $n$ ,prove that$$a_{n}\ge \frac{4}{\sqrt{9n+7}}.$$

Find the greatest positive integer $m$ , such that one of the $4$ letters $C,G,M,O$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.

In acute triangle $ABC, AB<AC$ , $I$ is its incenter, $D$ is the foot of perpendicular from $I$ to $BC$ , altitude $AH$ meets $BI,CI$ at $P,Q$ respectively. Let $O$ be the circumcenter of $\triangle IPQ$ , extend $AO$ to meet $BC$ at $L$ . Circumcircle of $\triangle AIL$ meets $BC$ again at $N$ . Prove that $\frac{BD}{CD}=\frac{BN}{CN}$ .

Attachments:

Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$ , define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$ For segment $MN$ , define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$ .

Find the smallest real number $\lambda$ , such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$ , satisfying: for any two points $M,N$ on $l$ , $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$