# Problem 587: Concave triangle
A square is drawn around a circle as shown in the diagram below on the
left. We shall call the blue shaded region the L-section. A line is
drawn from the bottom left of the square to the top right as shown in
the diagram on the right. We shall call the orange shaded region a
concave triangle. It should be clear that the concave triangle occupies
exactly half of the L-section. Two circles are placed next to each other
horizontally, a rectangle is drawn around both circles, and a line is
drawn from the bottom left to the top right as shown in the diagram
below. This time the concave triangle occupies approximately 36.46% of
the L-section. If n circles are placed next to each other horizontally,
a rectangle is drawn around the n circles, and a line is drawn from the
bottom left to the top right, then it can be shown that the least value
of n for which the concave triangle occupies less than 10% of the
L-section is n = 15. What is the least value of n for which the concave
triangle occupies less than 0.1% of the L-section?