Solutions to Problems 852-860

Q.852 If $a_1,a_2,\cdots a_n$ are positive real numbers and $a_1+a_2+\cdots + a_n =1$ prove that $$ \sum_{k=1}^n \left(a_k + \frac{1}{a_k}\right)^2 \leq \frac{(n^2+1)^2}{n}.$$