USAMO

[Mojo-Mojo]

$a,b,c,d \in \Re$
prove that $\sqrt{a^2+b^2}+\sqrt{c^2+d^2}\geqslant 2\sqrt{ac+bd} $

[nooonuii]

$\sqrt{ac+bd}\leq \sqrt{(a^2+b^2)(c^2+d^2)}\leq \dfrac{\sqrt{a^2+b^2}+\sqrt{c^2+d^2}}{2}$