HMMT-Nov 2015 guts, Problem 6

Let $A B$ be a segment of length 2 with midpoint $M$. Consider the circle with center $O$ and radius $r$ that is externally tangent to the circles with diameters $A M$ and $B M$ and internally tangent to the circle with diameter $A B$. Determine the value of $r$.

Solution: Let $X$ be the midpoint of segment $A M$. Note that $O M \perp M X$ and that $M X=\frac{1}{2}$ and $O X=\frac{1}{2}+r$ and $O M=1-r$. Therefore by the Pythagorean theorem, we have \[ O M^{2}+M X^{2}=O X^{2} \Longrightarrow(1-r)^{2}+\frac{1}{2^{2}}=\left(\frac{1}{2}+r\right)^{2} \] which we can easily solve to find that $r=\frac{1}{3}$. $\fbox{\frac{1}{3}}$.

Answer: \frac{1}{3}