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*}