Easy inequality
Source: 2020 Taiwan TST
March 28, 2020
InequalityinequalitiesTaiwan
Problem Statement
Let , , , 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*}