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.
Contents
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
Claim
Points and
are collinear.
Proof
is the midpoint of arc
Denote
Therefore
points
and
are collinear.
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One-to-one mapping of the circle
Let a circle two fixed points
and
on it and a point
inside it be given.
Then there is a one-to-one mapping of the circle
onto itself, based on the following two theorems.
1. Let a circle two fixed points
and
on
and a point
inside
be given.
Let an arbitrary point be given.
Let is the midpoint of the arc
Denote Prove that
2. Let a circle two fixed points
and
on
and a point
inside
be given.
Let an arbitrary point be given.
Let is the midpoint of the arc
Denote
Denote Prove that
Proof
Points are collinear.
2. Points and
are collinear (see Claim in 2024, Problem 23).
We use Pascal's theorem for points and crosspoints
and get
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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
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2024, Problem 21
A chord of the circumcircle of a triangle
meets the sides
at points
respectively. The tangents to the circumcircle at
and
meet at point
and the tangents at points
and
meets at point
The line
meets
at point
Prove that the lines and
concur.
Proof
WLOG,
Denote
Point is inside
We use Pascal’s theorem for quadrilateral and get
We use projective transformation which maps to a circle and that maps the point
to its center.
From this point we use the same letters for the results of mapping. Therefore the segments and
are the diameters of
is the midpoint
preimage lies on preimage
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2024, Problem 19
A triangle its circumcircle
, and its incenter
are drawn on the plane.
Construct the circumcenter of
using only a ruler.
Solution
We successively construct:
- the midpoint of the arc
- the midpoint of the arc
- the polar of point
- the polar of point
- the polar of the line
- the tangent to
- the tangent to
- the trapezium
- the point
- the point
- the midpoint of the segment
- the midpoint of the segment
- the diameter of
- the diameter of
- the circumcenter
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2024, Problem 18
Let be the altitudes of an acute-angled triangle
be its excenter corresponding to
be the reflection of
about the line
Points
are defined similarly. Prove that the lines
concur.
Proof
Denote the incenter of
Points
are collinear.
We will prove that
Denote
- semiperimeter.
The area
Points
are collinear, so the lines
concur at the point
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2024, Problem 16
Let and
be the bisectors of a triangle
The segments and
meet at point
Let
be the projection of
to
Points and
on the sides
and
respectively, are such that
Prove that
Proof
is the common side)
is the midpoint
is the midpoint of
(see Division of bisector for details.)
So Denote
Another solution see 2024_Sharygin_olimpiad_Problem_16
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2024, Problem 12
The bisectors of a
with
meet at point
The circumcircles of triangles meet at point
Prove that the line bisects the side
Proof
Denote the midpoint
In triangles
and
, by applying the law of sines, we get
We use the formulas for circle and get
In triangles
and
, by applying the law of sines, we get
Therefore The function
increases monotonically on the interval
This means and points
and
are collinear.
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2024, Problem 9
Let be a trapezoid circumscribed around a circle
centered at
which touches the sides
and
at points
respectively.
The line passing trough and parallel to the bases of trapezoid meets
at point
Prove that and
concur.
Solution
Solution 1.
is the center of similarity of triangles
and
Solution 2.
Denote
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2024, Problem 8
Let be a quadrilateral with
and
The incircle of touches the sides
and
at points
and
respectively.
The midpoints of segments and
are points
Prove that points are concyclic.
Solution
is the rotation of
around a point
through an angle
is the rotation of
around a point
through an angle
So is the rotation of
around a point
through an angle
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2024, Problem 2
Three distinct collinear points are given. Construct the isosceles triangles such that these points are their circumcenter, incenter and excenter (in some order).
Solution
Let be the midpoint of the segment connecting the incenter and excenter. It is known that point
belong the circumcircle.
Construction is possible if a circle with diameter IE (incenter – excenter) intersects a circle with radius OM (circumcenter – M). Situation when
between
and
is impossible.
Denote points such that
and
Suppose point is circumcenter, so
is incenter.
is midpoint BC. The vertices of the desired triangle are located at the intersection of a circle with center
and radius
with
and a line
Suppose point is circumcenter, so
is incenter.
is midpoint
The vertices of the desired triangle are located at the intersection of a circle with center
and radius
with
and a line
Suppose point is circumcenter, so
is incenter.
is midpoint
Suppose
The vertices of the desired triangle are located at the intersection of a circle with center
and radius
with
and a line
If there is not desired triangle.
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