# Problem 31

**Algebra**Level 5

\[a+b+c+d\geq \frac{m}{n}(ab+ac+ad+bc+bd+cd)\]

Given that \(a,b,c,d\) are non-negative and \(2(ab+ac+ad+bc+bd+cd)+abc+abd+acd+bcd=16\).Let \(\frac{m}{n}\) be the maximum value for which inequality holds and \(m,n\) are coprime integers. Compute \(m+n\).