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If $$0 < a \le x \le b$$, $$0 < a \le y \le b$$ and $$\frac{x}{b} = \frac{2x + y}{x + a + b}$$, then [ ]
Cross-multiplying gives $$x(x + a + b) = b(2x + y)$$, which rearranges to $$x(x - b) + (ax - by) = 0$$. Since $$0 < x \le b$$, the term $$x(x - b) \le 0$$; since $$a \le y$$ and $$x \le b$$ with all quantities positive, $$ax \le by$$, so $$ax - by \le 0$$. Two non-positive terms add to zero only if both are zero, so $$x = b$$ and then $$ab = by$$, giving $$y = a$$. Hence $$a = y \le x = b$$, with equality only when $$a = b$$.
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