Calvin draws 100 lines in the plane such that no two are parallel.
For each equilateral triangle formed by 3 of the lines, he receives $3. For each non-equilateral isosceles triangle, he receives $1.
What is the maximum amount of money Calvin can profit from this venture? (Answer in dollars)
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
I will give a generalised proof for n ≥ 3 lines.
Define a based isosceles triangle to be a triangle formed by three of the lines, with two sides of equal length marked. Then the amount of money obtained by Calvin is equal to the number of based isosceles triangles.
For each ordered pair of distinct lines ( l 1 , l 2 ) , there is at most one isosceles triangle with l 1 unmarked and l 2 marked. So there are at most ⌊ 2 n − 1 ⌋ based isosceles triangles with l 1 unmarked. Therefore, there are at most n × ⌊ 2 n − 1 ⌋ based isosceles triangles altogether.
If n is odd, Calvin can achieve this upper bound by drawing the lines to form the sides of a regular polygon with n sides. If n is even, Calvin can achieve this upper bound by drawing the lines to form all but one of the sides of a regular polygon with n + 1 sides.
Substituting n = 1 0 0 , we get Calvin profits $4900.