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This question is flawed.

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It can be shown using AM-GM that the altitude to the hypotenuse is at most half of the hypotenuse. It follows that there is no such triangle, and hence the correct answer is $\boxed{\text{This question is flawed}}$ .

Source:A post on MSETo quote @Calvin Lin,

...This is one reason why I stress considering the possibility of the geometrical setup and ensuring that it actually exists.The main point of checking the existence of the possibility of a geometric structure isn't necessarily that the question is deliberately trying to fool you (like this one), but that checking

whya geometric structure exists will give you a pointer towards some narrow but definite parts of a diagram that have more restrictions imposed on them (otherwise the diagram would cease to exist). These parts of a diagram should theoretically be easier to investigate due to having more restrictions on them. This is still true in olympiad maths, not just pre-olympiads.