Sharygin Olympiads, the best
Igor Fedorovich Sharygin (13/02/1937 - 12/03/2004, Moscow) - Soviet and Russian mathematician and teacher, specialist in elementary geometry, popularizer of science. He wrote many textbooks on geometry and created a number of beautiful problems. He headed the mathematics section of the Russian Soros Olympiads. After his death, Russia annually hosts the Geometry Olympiad for high school students. It consists of two rounds – correspondence and final. The correspondence round lasts 3 months.
The best problems of these Olympiads will be published. The numbering contains the year of the Olympiad and the serial number of the problem. Solutions are often different from the original ones.
2024, Problem 23
A point moves along a circle
Let
and
be fixed points of
and
be an arbitrary point inside
The common external tangents to the circumcircles of triangles and
meet at point
Prove that all points lie on two fixed lines.
Solution
Denote
is the circumcenter of
is the circumcenter of
Let and
be the midpoints of the arcs
of
Let and
be the midpoints of the arcs
of
These points not depends from position of point
Suppose, see diagram).
Let
Similarly,
Let
Therefore Similarly, if
then
vladimir.shelomovskii@gmail.com, vvsss
2024, Problem 22
A segment is given. Let
be an arbitrary point of the perpendicular bisector to
be the point on the circumcircle of
opposite to
and an ellipse centered at
touche
Find the locus of touching points of the ellipse with the line
Solution
Denote the midpoint
the point on the line
In order to find the ordinate of point
we perform an affine transformation (compression along axis
which will transform the ellipse
into a circle with diameter
The tangent of the
maps into the tangent of the
Denote
So point is the fixed point (
not depends from angle
)
Therefore point
lies on the circle with diameter
(except points
and
vladimir.shelomovskii@gmail.com, vvsss