CWMO 2004

All the grids of a $m\times n$ chess board ( $m,n\geq 3$ ), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by some specific color grids? Justify your answers.

Find all reals $k$ such that $$a^3 + b^3 + c^3 + d^3 + 1\geq k(a + b + c + d)$$ holds for all $a,b,c,d\geq - 1$ .

Edited by orl.