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Putnam 1977 A6

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April 7, 2022
college contests

Problem Statement

Let f(x,y)f(x,y) be a continuous function on the square S={(x,y):0≤x≤1,0≤y≤1}.S=\{(x,y):0\leq x\leq 1, 0\leq y\leq 1\}. For each point (a,b)(a,b) in the interior of SS, let S(a,b)S_{(a,b)} be the largest square that is contained in SS, is centered at (a,b)(a,b), and has sides parallel to those of SS. If the double integral ∫∫f(x,y)dxdy\int \int f(x,y) dx dy is zero when taken over each square S(a,b)S_{(a,b)}, must f(x,y)f(x,y) be identically zero on SS?
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