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Let $x_1, x_2, y_1, y_2, z_1, z_2$ be non-negative real numbers and $a_1, a_2, b_1, b_2$ be positive integers such that $$\frac{x_1}{a_1} \geq \frac{y_1}{1} \geq \frac{z_1}{b_1} \ \text{ and } \ \frac{x_2}{a_2} \geq \frac{y_2}{1} \geq \frac{z_2}{b_2}.$$ It is also given that $a_1 + b_1 = a_2 + b_2$ and $z_1 \geq z_2$.

Is the following inequality always true? $$\frac{x_1 + x_2 + y_1}{a_1 + a_2 + 1} \geq \frac{z_1 + z_2 + y_2}{b_1 + b_2 + 1}$$

Note that if $z_1 \geq z_2$ is not given as a condition, then the inequality is false: a counterexample is such that $$\frac{2}{4} \geq \frac{0.5}{1} \geq \frac{1}{2} \ \text{ and } \ \frac{1}{1} \geq \frac{1}{1} \geq \frac{5}{5},$$ but $$\frac{2 + 1 + 0.5}{4 + 1 + 1} < \frac{1 + 5 + 1}{2 + 5 + 1}.$$

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    I posted an in correct answer and then deleted. Where is the motivation of this come from ? – dezdichado Nov 21 '23 at 18:11
  • @dezdichado It is from a somewhat unrelated research question, where having some insights to the answer to this question will help. –  Nov 21 '23 at 22:41

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Here is a counterexample: $$\frac31\ge\frac31\ge\frac93,\qquad\frac33\ge\frac11\ge\frac11,$$ $$1+3=3+1, \ \text{ and } \ 9\ge1$$ but $$\frac{3+3+3}{1+3+1}<\frac{9+1+1}{3+1+1}.$$

Anne Bauval
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