The USSR Olympiad Problem Book
#258. Prove that if \(a+b=1\), where \(a\) and \(b\) are positive numbers, then: \begin{align*} \left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}.\\ \end{align*}
Hint. Use: \begin{align*} A^2 + B^2 &\geq 2 \cdot \left(\frac{A+B}{2}\right)^2. \end{align*}