Question 386579
Here is a nice geometry proof question:


ABC is a triangle with angle A = 90. Points D and E lie on sides AC and AB, respectively, such that angle ABD = angle DBC and angle ACE = angle ECB. Segments BD and CE meet at I. Determine whether or not it is possible for segments AB, AC, BI, ID, CI, and IE to all have integer side lengths (2010 USAMO, #1).


There are probably multiple ways to solve this; my solution during the actual exam involved a little trigonometry. I encourage you to go for it and challenge yourself!