A Pythagorean Denominator Bound

Problem

Let $a, b, c > 0$ with $a^2 + b^2 = c^2$ . Find the largest real number $k$ such that

\frac{1}{32a} + \frac{1}{32b} + \frac{1}{c} \geqslant \frac{k}{a + b + c}

holds for all such $a, b, c$ .

Answer

Solution

Difficulty	7/10
Topics	Substitution, AM-GM, inequality

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