Exercise 1627Combinatorics › Counting principlesOlympiadAdded 3 Nov 2026

Pigeonhole with points

#pigeonhole#geometry#Dirichlet

Statement

Show that among any $5$ points in a unit square, two are at distance $\le\sqrt2/2$ .

Hint

Divide the square into $4$ subsquares of side $1/2$ .

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Subsquares. Divide the unit square into $4$ subsquares of side $1/2$ each.

Pigeonhole. With $5$ points and $4$ subsquares, some subsquare contains at least $2$ points.

Distance. The maximum distance between two points in a subsquare of side $1/2$ is $\sqrt{(1/2)^2+(1/2)^2}=\frac{\sqrt2}{2}$ .

Conclusion. Two points are at distance $\le\sqrt2/2$ .