A computer programmer claims that he generated six real numbers a_1, a_2,...,a_6 so that the sum of any four consecutive a_i is positive,
but the sum of any 3 consecutive a; is negative. Prove that his claim
is false.
$\displaystyle a_1+a_2+a_3+a_4>0 $ and $\displaystyle a_1+a_2+a_3<0$ therefore $\displaystyle a_4>0$. Following this idea you get $\displaystyle a_5>0$ and $\displaystyle a_6>0$. But then how can $\displaystyle a_4+a_5+a_6$ be negative?!