CN-LEAGUE 2021

Let $k\ge 2$ be an integer and $a_1,a_2,\cdots,a_k$ be $k$ non-zero reals. Prove that there are finitely many pairs of pairwise distinct positive integers $(n_1,n_2,\cdots,n_k)$ such that $$a_1\cdot n_1!+a_2\cdot n_2!+\cdots+a_k\cdot n_k!=0.$$

In $\triangle ABC$ , point $M$ is the middle point of $AC$ . $MD//AB$ and meet the tangent of $A$ to $\odot(ABC)$ at point $D$ . Point $E$ is in $AD$ and point $A$ is the middle point of $DE$ . $\{P\}=\odot(ABE)\cap AC,\{Q\}=\odot(ADP)\cap DM$ . Prove that $\angle QCB=\angle BAC$ .

If $n\ge 4,\ n\in\mathbb{N^*},\ n\mid (2^n-2)$ . Prove that $\frac{2^n-2}{n}$ is not a prime number.

Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$ , if $|A| >cn$ , there exists a function $f:A\to\{1,-1\}$ satisfying $$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$