A Summer Problem

Since the Problem of the Month is on summer holiday, we thought we’d challenge everyone to a problem out of the Pi Mu Epsilon Journal problem section:

Given a,b,c,d in the interval [0,1] such that at most one of them is equal to zero, prove that

\frac{1}{a^2+b^2}+\frac{1}{b^2+c^2}+\frac{1}{c^2+d^2}+\frac{1}{d^2+a^2} \geq \frac{8}{3+abcd}

— proposed by Tuan Le, student at Fairmont High School in Fairmont, CA

Good luck!