Question 22
(KOR 2) Let $p$ be the product of two consecutive integers greater than $2$. Show that there are no integers $x_1,x_2,\ldots,x_p$ satisfying the equation $$\sum_{i=1}^p x_i^2-\frac4{4p+1}\left(\sum_{i=1}^p x_i\right)^2=1.$$ Alternative formulation. Show that there are only two values of $p$ for which there are integers $x_1,x_2,\ldots,x_p$ satisfying the above inequality.