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Problem 876

AMC 12 late, AIME early

Combinatorics Difficulty 4.7 Prove it Brazilian Math Olympiad · Brazil

Consider 1000 points inside a square with sidelength 16. Prove that there is an equilateral triangle with sidelength $2\sqrt{3}$ that covers at least 16 of those points.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Since $\left(\frac{16}{2\sqrt{3}}\right)^2 = \frac{64}{3} = 21+\frac{1}{3}$ lies between $4.5^2 = 20.25$ and $5^2$ and the altitude of the triangle is $\frac{2\sqrt{3}\cdot\sqrt{3}}{2} = 3$ , we can cover a square with sidelength $16$ with $2 \cdot 5 \cdot \lfloor\frac{16}{3}\rfloor = 60$ equilateral triangles. Since $\lfloor\frac{1000}{60}\rfloor = 16$ , by the pigeon hole principle there is an equilateral triangle that covers at least $17$ points.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.