Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 699. (May 2017)

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 pont)

Deadline expired on June 12, 2017.

### Statistics:

 3 students sent a solution. 5 points: Williams Kada. 4 points: Bukva Balázs. 2 points: 1 student.

Problems in Mathematics of KöMaL, May 2017