KVPY SA Exam 05-Nov-2017 Question Paper

Let ${\mathit{a}}_1,{\mathit{a}}_2,....{\mathit{a}}_{\mathit{n}}$ be n nonzero real numbers, of which p are positive and remaining are negative. The number of ordered pairs (j, k), j < k, for which ${\mathit{a}}_{\mathit{j}}{\mathit{a}}_{\mathit{k}}$ is positive, is 55. Similarly, the number of ordered pairs (j, k), j < k, for which ${\mathit{a}}_{\mathit{j}}{\mathit{a}}_{\mathit{k}}$ is negative is 50. Then the value of ${\mathit{p}}^2+{\left(\mathit{n}-\mathit{p}\right)}^2$ is