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The number of pairs $$(f(x), g(y))$$ of functions such that $$x^2 + xy + y^2 = f(x) + g(y)$$ for all $$x, y \in \mathbb{R}$$ is _____
Correct Answer: 0
For any $$f$$ and $$g$$, the combination $$h(1, 1) - h(1, 0) - h(0, 1) + h(0, 0)$$ of $$h(x, y) = f(x) + g(y)$$ is always 0. For $$x^2 + xy + y^2$$ the same combination is $$3 - 1 - 1 + 0 = 1 \ne 0$$. So no such pair of functions exists, and the number of pairs is 0.
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