Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

5. Given $x, y \in \mathbf{R}_{+}$ . Then the maximum value of $\frac{x}{2 x+y}+\frac{y}{x+2 y}$ is ( ).

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

5. B.

Let $s=2x+y, t=x+2y$ . Then
$\begin{array}{l} x=\frac{1}{3}(2s-t), \\ y=\frac{1}{3}(2t-s) . \end{array}$
$\text{Thus } \frac{x}{2x+y}+\frac{y}{x+2y}=\frac{4}{3}-\frac{1}{3}\left(\frac{t}{s}+\frac{s}{t}\right) \leqslant \frac{2}{3} \text{. }$

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