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Given a square with sidelenght $\sqrt{2019}$ partitioned into finite number of rectangles one needs to show that at least one of them must have both sides irrational. It's obviously one of them must have one irrational side, but after I'm stuck. (Problem from MSU student competetion 2019)

Anton Shcherbina
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  • HINT.-I think that whatever the (finite!) partition is we can do such that we get two finite sums of horizontal and vertical segments such that $$\sum h_i=\sqrt{2019}=\sum v_j$$. From this the conclusión. – Ataulfo Jun 13 '24 at 21:23
  • @Piquito how conclusion possible? – Anton Shcherbina Jun 13 '24 at 21:32
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    Hint: integrate the function $\sin(2k\pi x)\sin(2k\pi y)$, where $k$ is a common multiple of the denominators of each of the rational side lengths of the rectangles. – Dan Rust Jun 13 '24 at 23:02
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    Another approach: show that if each rectangle has at least one pair of rational edges, then there exists a path from one corner of the square to another corner following only rational edges in the partition. – Dan Rust Jun 13 '24 at 23:08
  • @Anton Shcherbina: In the first (or second) sum at least one of addends should be irrational. – Ataulfo Jun 14 '24 at 00:09

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Assume there exists a partition such that every rectangle has at least one rational side. Let $N$ be the product of the denominators of all such rational side lengths. Scaling the entire square by $N$, we have a square with sides $\sqrt{2019}N$ partitioned into rectangles with at least one integer side each. This is a relatively well-known problem, and you can find a plethora of solutions, for example at this mse question and the links there.

For completeness, I'll include a short proof here: Colour the square in a checkerboard pattern with tiles of size $\frac12\times\frac12$. The key insight is then that a grid-aligned rectangle has at least one integer side if and only if it contains the same area of black and white colouring. Therefore, the black and white areas are equal in each rectangle of the partition, but not in the whole square, a contradiction.

anankElpis
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