$\begin{cases} xyzw & = 1 \\ x^{100} + 100y & = 100z + w^{100} \\ 100x + y^{100} & = z^{100} + 100w \end{cases}$

Let $(x,y,z,w)$ be positive real numbers that satisfy the system of equations above. Find the minimum possible value of $x + y + z + w$ .

For more problems on finding maximum and minimum, click here .

The answer is 4.

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As $x, y, z, w$ are positive real numbers, we have that $\frac { x+y+z+w }{ 4 } \ge \sqrt [ 4 ]{ xyzw } =1$ . Thus $min(x + y + z + w) = 4$ .

By the way, the second and third equation is somewhat useless.