Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

6. (15 points) On a piece of acute triangle scrap, process a square part so that the four vertices of the square are all on the edges of the triangle. If the three sides of the triangle are $a$ , $b$ , and $c$ ( $a > b > c$ ), try to find: On which side should the two vertices of the square be placed to maximize the area of the square part? Prove your conclusion.

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Solution

6. Let the side lengths of the squares on $a$ , $b$ , and $c$ be $x_{a}$ , $x_{b}$ , and $x_{c}$ respectively (Figure 8 shows a scenario where two vertices of the square lie on the edge $a$ ), and the heights on $a$ , $b$ , and $c$ are denoted as $h_{a}$ , $h_{b}$ , and $h_{c}$ . Then $h_{a}<h_{b}<h_{c}$ . Hence

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