Let $p$ and $q$ be positive numbers having a sum of $1.$ What is the minimum value of $(p+(1/p))^2 + (q+(1/q))^2?$
Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$