Baltic Way 1994 · Problem 16

Solution:

It suffices to prove that if the distance between the centres of two Hedgehogs is less than $0.2$, then these Hedgehogs intersect. To show this, consider two Hedgehogs with their centres at points $O$ and $M$, respectively, such that $|OM| < 0.2$. Let $A$, $B$ and $C$ be the endpoints of the needles of the first Hedgehog (see Figure 4) and draw a straight line $l$ parallel to $AC$ through the point $M$. As $|AC| = \sqrt{3}$ implies $|KL| \leq \frac{0.2}{0.5}|AC| < 1$ and the second Hedgehog has at least one of its needles pointing inside the triangle $OKL$, this needle intersects the first Hedgehog.

Figure 4