CN-TST 2009

Let positive real numbers $a,b$ satisfy $b - a > 2.$ Prove that for any two distinct integers $m,n$ belonging to $[a,b),$ there always exists non-empty set $S$ consisting of certain integers belonging to $[ab,(a + 1)(b + 1))$ such that $\frac { \prod_{x\in S}}{mn}$ is square of a rational number.

Let $m > 1$ be an integer, $n$ is an odd number satisfying $3\le n < 2m,$ number $a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $(1)$ for any $1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $1,2,3,\cdots,m; (2)$ for any $1 < i\le m, 1\le j\le n - 1, |a_{i,j} - a_{i,{j + 1}}|\le 1$ holds. Find the minimal value of $M$ , where $M = max_{1 < i < m}\sum_{j = 1}^n{a_{i,j}}.$

Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $2^m + 3^n$ or not. where $m,n$ are nonnegative integers.