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Solution: Let the sides be $ a $ and $ b $. By the Pythagorean theorem, $ a^2 + b^2 = 25^2 = 625 $. The area $ A = ab $. To maximize $ A $, use the identity $ (a + b)^2 = a^2 + 2ab + b^2 $, but it is more direct to note that for fixed $ a^2 + b^2 $, $ ab $ is maximized when $ a = b $. Thus, $ 2a^2 = 625 \Rightarrow a = rac{25}{\sqrt{2}} $. The maximum area is $ A = \left( rac{25}{\sqrt{2}}
ight)^2 = rac{625}{2} = 312.5 $. oxed{\dfrac{625}{2}}
Question: Find the number of functions $ f : \mathbb{R} o \mathbb{R} $ such that $ f(a + b) = f(a) + f(b) + ab $ for all real $ a, b $.

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