# Victor's Level 5 Challenges - 3

Algebra Level 5

There exists a constant $$C$$ such that

$\sum_{i=1}^4\left(x_i+\frac{1}{x_i}\right)^3 \geq C$

for all positive real numbers $$x_1,x_2,x_3,x_4$$ which satisfy

$x_1^3+x_3^3+3x_1x_3=x_2+x_4=1.$

Given that the largest constant $$C$$ can be expressed in the form $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, positive integers, find the value of $$m+n$$.

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