**A. 699.** A circle \(\displaystyle \omega\) lies in a circle \(\displaystyle \Omega\) such that their common center is the point \(\displaystyle O\). Fix a point \(\displaystyle A\ne O\) inside \(\displaystyle \omega\). Let \(\displaystyle X\) denote an arbitrary point on the circumference of \(\displaystyle \Omega\), and let \(\displaystyle Y\) denote the second intersection point of \(\displaystyle \Omega\) and the line \(\displaystyle AX\). Let \(\displaystyle Z\) denote the intersection of \(\displaystyle \omega\) and the line segment \(\displaystyle AX\). Let \(\displaystyle M\) denote the point on the line segment \(\displaystyle AZ\) for which \(\displaystyle MX\cdot MZ\cdot AY = MA\cdot MY\cdot XZ\). Let \(\displaystyle x\) and \(\displaystyle y\) denote the tangents of the circle \(\displaystyle \Omega\) at the points \(\displaystyle X\) and \(\displaystyle Y\), respectively. Let \(\displaystyle t\) denote the line that passes through \(\displaystyle M\) and either also passes through the intersection of \(\displaystyle x\) and \(\displaystyle y\), or is parallel to both \(\displaystyle x\) and \(\displaystyle y\). Finally, let \(\displaystyle T\) denote the intersection of \(\displaystyle t\) and the line \(\displaystyle OZ\).

Show that the locus of the points \(\displaystyle T\), as \(\displaystyle X\) is varied, is an ellipse, and the line \(\displaystyle t\) is a tangent of this ellipse.

(5 points)

**Deadline expired on 12 June 2017.**