2025 SSMO Accuracy Round Problems/Problem 3
Problem
Nonnegative real numbers 
and 
satisfy ![\[\frac{\sqrt{x}+13}{y} = \frac{\sqrt{y}+29}{z} = \frac{\sqrt{z} + 46}{x} = 2\]](http://latex.artofproblemsolving.com/5/e/f/5efb565a73ba272461c8e651f42057f2e54531fd.png)
and ![\[\frac{\sqrt{x} + \sqrt{y}+\sqrt{z}}{x+y+z} = \frac{6}{25}.\]](http://latex.artofproblemsolving.com/c/3/c/c3cb7e8a68d444b529aabbfdd5782dbbd9b46f76.png)
Find the value of 
.
Solution
From the first chain of equalities in the problem statement, when we clear any appearing fractions, we have
\begin{align*}
\sqrt{x} &= 2y - 13 \\
\sqrt{y} &= 2z - 29 \\
\sqrt{z} &= 2x - 46. \\
\end{align*}
Adding these three equations together, we obtain
![\[\sqrt{x}+\sqrt{y}+\sqrt{z} = 2(x+y+z) - 88.\]](http://latex.artofproblemsolving.com/a/6/5/a65881c377e4e6174d44496098b321dbef5724ec.png)
Plugging this into the last equation in the problem statement, we have
![\[\frac{2(x+y+z)-88}{x+y+z} = \frac{6}{25}.\]](http://latex.artofproblemsolving.com/f/f/2/ff27ab4c044c865058fde14de2faa9394e804cac.png)
It is straightforward to solve this equation for ; we find that 
.
~Sedro
