KMO 2015 Problem 4

For positive integers $n, k, l$ , we define the number of $l$ -tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$ .

(i): $n=a_1+a_2+\cdots +a_l$ (ii): $a_1>a_2>\cdots > a_l > 0$ .

(iii): $a_l$ is an odd number.

(iv): There are $k$ odd numbers out of $a_i$ .

For example, from $9=8+1=6+3=6+2+1$ , we have $Q(9,1,1)=1$ , $Q(9,1,2)=2$ , $Q(9,1,3)=1$ .

Prove that if $n>k^2$ , $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.