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Difference between revisions of "2025 SSMO Accuracy Round Problems/Problem 3"

(Created page with "==Problem== Nonnegative real numbers <math>x,y,</math> and <math>z</math> satisfy <cmath>\frac{\sqrt{x}+13}{y} = \frac{\sqrt{y}+29}{z} = \frac{\sqrt{z} + 46}{x} = 2</cmath> a...")
 
(→‎Solution)
 
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
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 +
From the first chain of equalities in the problem statement, when we clear any appearing fractions, we have
 +
\begin{align*}
 +
\sqrt{x} &= 2y - 13 \\
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\sqrt{y} &= 2z - 29 \\
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\sqrt{z} &= 2x - 46. \\
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\end{align*}
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Adding these three equations together, we obtain
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<cmath>\sqrt{x}+\sqrt{y}+\sqrt{z} = 2(x+y+z) - 88.</cmath>
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Plugging this into the last equation in the problem statement, we have
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<cmath>\frac{2(x+y+z)-88}{x+y+z} = \frac{6}{25}.</cmath>
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It is straightforward to solve this equation for <math>x+y+z</math>; we find that <math>x+y+z = \boxed{50}</math>.
 +
 +
~Sedro

Latest revision as of 02:43, 11 September 2025

Problem

Nonnegative real numbers $x,y,$ and $z$ satisfy \[\frac{\sqrt{x}+13}{y} = \frac{\sqrt{y}+29}{z} = \frac{\sqrt{z} + 46}{x} = 2\] and \[\frac{\sqrt{x} + \sqrt{y}+\sqrt{z}}{x+y+z} = \frac{6}{25}.\] Find the value of $x+y+z$.

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 \[\sqrt{x}+\sqrt{y}+\sqrt{z} = 2(x+y+z) - 88.\] Plugging this into the last equation in the problem statement, we have \[\frac{2(x+y+z)-88}{x+y+z} = \frac{6}{25}.\] It is straightforward to solve this equation for $x+y+z$; we find that $x+y+z = \boxed{50}$.

~Sedro

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