Putnam and Beyond prob 117

1. May 15, 2008

ehrenfest

1. The problem statement, all variables and given/known data
Let ABCD be a convex cyclic quadrilateral. Prove that

$$|AB-CD|+|AD-BC| \geq 2|AC-BD|$$

2. Relevant equations

3. The attempt at a solution
First, isn't a cyclic quadrilateral always convex?

2. May 16, 2008

tiny-tim

Hi ehrenfest!

Yes … "convex" seems unnecessary! :yuck:

3. May 16, 2008

DavidWhitbeck

Putnam is supposed to be for fun Ehrenfest. If you ask for help on every problem that you can't immediately solve... how are you having fun? The pleasure is all in finding the aha! moment yourself.

4. May 16, 2008

ehrenfest

So, I can get triangle inequalities for the triangles ABC, BCD, ACD, ABD. Put there are 12 of them and I tried to play around with so they would look similar to the inequality in the problem statement but I did not get very far.