ILL 1992 Problem 49

Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that $$\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m ( \ x_1=x_{4k+3} \ )$$ prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that $$\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m ( \ x_{k_{1}} = x_{k_{7}} \ )$$