CN-LEAGUE 2009

Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$ , intersecting $\omega$ at $P$ (different from $C$ ). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$ .

1. Prove that $MP\cdot MT=NP\cdot NT$ ;
2. Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$ . Prove that $Q,I,J,T$ are concyclic.

Let $n$ be a positive integer. Prove that $$-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}$$

Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$ .

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$ . The following conditions are given:

• Every row consists of distinct numbers;
• $\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$ ;
• $x_{17}=x_{28}=x_{39}=0$ ;
• $x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$ .
• The first three columns of $P$ satisfy the following property $(R)$ : for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$ , $1\le k\le 9$ , there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$ .

Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$ , where $l\ge 4$ , satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$ , then this column uniquely exists.