Problem

Source: PMO 2021

Tags: algebra, number theory, inequalities, Philippines, PMO



Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$$$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$.