Re: Two Olympiad Problems
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You don't have to use trigonometry to prove that the height is maximized when the triangle is isoceles. You can use the triangle inequality after adding a reflected copy of the triangle above. So, this whole problem can be done without calculus.
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All calculus problems can be done without calculus. When my father was teaching mathematical logic at City College he would sometimes ask for a calculus problem and demonstrate.
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Two comments:
1) I've never understood the resistance to using calculus on these types of problems, as if it is some kind of failure. Finding different roads to a solution is often very interesting, but what's wrong with calculus?
2) All calculus problems? How would one frame a generic differential equation in language that does not involve calculus? Is there some kind of geometric trickery that will give me Bessel functions?
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