In a disk of radius 10, how can we find all values n such that there are exactly n points in the disk and such that no matter how the n points are arranged, we can draw a disk with radius 1 in the disk of radius 10 such that the small disk does not contain any of the n points? (points may be on the circumference of the small circle but cannot be strictly inside, if that distinction makes a difference)
2026-03-28 04:33:38.1774672418
pigeonhole principle on a circle
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Hint:Check for the biggest square that can fit in the circle.The biggest square has $\sqrt 200 =14,14$ side.So there are $14*14$ squares of side $1$ in it.Now there are $15*15=225$ vertexs of the squares in the big disk...