# 2010 USAMO Problems/Problem 4

## Problem

Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments to all have integer lengths.

## Solution

We know that angle , as the other two angles in triangle add to 45^{\circ}AB, BC, BICIBC^2 = BI^2 + CI^2 - 2BI*CI*cos 135^{\circ}BC^2 = AB^2 + AC^2cos 135^{\circ} = -\frac{\sqrt{2}}{2}AB^2 + AC^2 - BI^2 - CI^2 = BI*CI*\sqrt{2}$$ (Error compiling LaTeX. ! Missing $ inserted.)\sqrt{2} = \frac{AB^2 + AC^2 - BI^2 - CI^2}{BI*CI}\sqrt{2}\sqrt{2}AB, BC, BICI$ to all be integers, which invalidates the original claim that all six lengths are integers, and we are done.