A rectangle is divided into smaller rectangles of random sizes each having at least one side of integer length. Prove that larger rectangle has at least one side of integral length.
I don't know how to approach this problem and where to start from.
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Let $f(x,y)=\sin(\pi x)\,\sin (\pi y)$ and note that $\int_{y_0}^{y_1}\int_{x_0}^{x_1} f(x,y)\,\mathrm dx\,\mathrm dy$ is zero iff at least one of $x_1-x_0$, $y_1-y_0$ is an integer.
Hagen von Eitzen
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This doesn't really hold, e.g. choose $y_1 = x_1 = 1$ and $y_0 = x_0 = 0$ – Richard Apr 29 '22 at 15:23
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In fact, one simply has to put $2\pi$ inside the function instead of just $\pi$ – Richard May 03 '22 at 11:45