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4 Variable Inequality

Source: Iran TST 2015 - second exam - first day problem 1

June 12, 2015

inequalities unsolvedinequalities

Problem Statement

a,b,c,d

are positive numbers such that

\sum_{cyc} \frac{1}{ab} =1

. Prove that :

abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}