Here is a puzzle I found in *Mitteilungen der DMV* (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch rabbits), and he claims that less than 5% of his subjects could solve it in under 1 hour. He tested it on students of mathematics and professors of mathematics, computer science, and engineering.

See if you have more luck. The problem is deceptively simple:

Suppose that you have a triangle $ABC$ and a point $D$ inside the triangle. Prove that the perimeter of $ABC$ is larger than the perimeter of $ABD$.

I am currently working on a generalization: Given two convex shapes $S$ and $T$, where $T$ totally encloses $S$. Prove that the perimeter of $S$ is no bigger than the perimeter of $T$.

Alternatively, for a shapes with straight edges: Prove that the perimeter of the convex hull of a set of points increases monotonically (but not strictly monotonically) as points are added to the set.

Please try to find an elementary proof for the special case of the triangle.

**Edit:** Thanks for all the nice answers. By now, I have found a really elementary proof on my own that just uses the triangle inequality twice.