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Tricky system of equations in the Centro

Source: Centroamerican and Caribbean Math Olympiad 2024 P5

October 18, 2024

algebrasystem of equationsOMCC

Problem Statement

Let

x

and

y

be positive real numbers satisfying the following system of equations:

\begin{cases} \sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\ \sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2 \end{cases}

Find the maximum value of

x + y