CN-LEAGUE 2013

Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$ , and for integers $m\ge 1$ ,

$$\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, a_{2m+1}=a_m+v, \end{cases} \end{array}$$

Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$ . Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$ .

Let $n,k$ be integers greater than $1$ , $n<2^k$ . Prove that there exist $2k$ integers none of which are divisible by $n$ , such that no matter how they are separated into two groups there exist some numbers all from the same group whose sum is divisible by $n$ .