Problem

Source: 2020 Taiwan TST

Tags: Inequality, inequalities, Taiwan



Let $a$, $b$, $c$, $d$ be real numbers satisfying \begin{align*} (a + c)(b + d) = \sqrt{2}(ac - 2bd - 1). \end{align*}Show that \begin{align*} (ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4. \end{align*}