Track / Stage 6 / 197 of 400 #1197 of 2000

Problem 1197

National olympiad, first round

Geometry Difficulty 6.2 Prove it

## PROBLEM 17. 0 - $k$ -CLOUDS

Circles of unit radius have no common interior points and lie within a strip $S$ , formed by two parallel lines, spaced apart by a distance $w$ .

Fig. 21

We will call these circles a $k$ -cloud if every line intersecting $S$ intersects at least $k$ circles.

Prove that for a 2-cloud, $w \geqslant 2+\sqrt{3}$ (Fig. 21).

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solution. Draw a line $m$ through point $O$ - the center of some circle $C$ from the 2-cloud, perpendicular to the lines bounding the strip $S$ (Fig. 22). Then the line $m$ must intersect another circle $A$ . Let $Q$ -

Fig. 22

be the foot of the perpendicular dropped from the center $P$ of circle $A$ to the line $m$ .

Since the line $m$ intersects $A$ , the length $P Q$ is no more than the radius. Therefore, $P Q \leqslant 1$ . And since circles $C$ and $A$ do not intersect, $O P \geqslant 2$ . By the Pythagorean theorem 28
we have that

$O Q=\sqrt{O P^{2}-P Q^{2}} \geqslant \sqrt{2^{2}-1^{2}}=\sqrt{3} .$

Since the strip $S$ must extend at least the radius of the circle on each side of the endpoints of the segment $O Q$ to contain circles $C$ and $A$ , its width

$w \geqslant 2+O Q=2+\sqrt{3}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.