2010 USAMO Problems/Problem 4

Problem

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$ . 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 $AB, AC, BI, ID, CI, IE$ to all have integer lengths.

Solution

We know that angle $BIC = 135^{\circ}$ , as the other two angles in triangle $BIC$ add to $45^{\circ}$ . Assume that only $AB, BC, BI$ , and $CI$ are integers. Using the Law of Cosines on triangle BIC,

$BC^2 = BI^2 + CI^2 - 2BI*CI*cos 135^{\circ}$ . Observing that $BC^2 = AB^2 + AC^2$ and that $cos 135^{\circ} = -\frac{\sqrt{2}}{2}$ , we have

$AB^2 + AC^2 - BI^2 - CI^2 = BI*CI*\sqrt{2}$

$\sqrt{2} = \frac{AB^2 + AC^2 - BI^2 - CI^2}{BI*CI}$

Since the right side of the equation is a rational number, the left side (i.e. $\sqrt{2}$ ) must also be rational. Obviously since $\sqrt{2}$ is irrational, this claim is false and we have a contradiction. Therefore, it is impossible for $AB, BC, BI$ , and $CI$ to all be integers, which invalidates the original claim that all six lengths are integers, and we are done.

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2010 USAMO Problems/Problem 4

Problem

Solution