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Is there a "Cauchy-Schwarz proof" of the following inequality? Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has $$ \int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...

The following is a holdover from my math contest days that I never got around to solve. We will use the notation $\left[ k\right] $ for the set $\left\{ 1,2,\ldots,k\right\} $ whenever $k$ is a ...