## Problem of the Day #8: A House with a ViewMarch 27, 2011

Posted by Saketh in : potd , trackback

A (very small) house is to be constructed at the center of a circular plot of land with radius $20$ meters. The developer plans to place trees around the house in a rather peculiar manner. Taking the house to be the origin, a sapling will be planted at every lattice point within the circle. The unit distance is to be $1$ meter.

Due to recent developments in genetic engineering, the trees can be modified to stop growing at any desired girth. To reduce construction costs, however, all of the trees will be identical.

The inhabitants of the house wish to be able to see outside of the plot. That is, there must exist some ray we can draw from the origin to the perimeter of the circle which does not intersect any of the trees. Determine the greatest $R$ such that for any tree radius $r < R$, the view is not blocked.

Assume for the purposes of this problem that the house is a single point at the origin.