We solve problems

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.

Problem #PRU-116301

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Area

13–14 4.0

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.

Problem #PRU-116573

Problems Geometry Plane geometry Circles Diameter, chords, and secants Chords and secants (other) Polygons Regular polygons

15–17 3.0

We create some segments in a regular \(n\)-gon by joining endpoints of the \(n\)-gon. What’s the maximum number of such segments while ensuring that no two segments are parallel? The segments are allowed to be sides of the \(n\)-gon - that is, joining adjacent vertices of the polygon.

Problem #PRU-21982

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

11–13 3.0

Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.

Problem #PRU-21983

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

51 points were thrown into a square of side 1 m. Prove that it is possible to cover some set of 3 points with a square of side 20 cm.

Problem #PRU-30290

Problems Number Theory Divisibility Odd and even numbers Geometry Plane geometry Polygons

10–12 2.0

a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.

b) What can be said about the case of a decagon?

Problem #PRU-32032

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality (other)

13–14 3.0

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

Problem #PRU-32038

Problems Methods Examples and counterexamples. Constructive proofs Geometry Plane geometry Lines, segments, rays, and angles Measurement of segments and angles. Adjacent angles.

11–13 2.0

Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.

Problem #PRU-32095

Problems Geometry Plane geometry Visual geometry

10–13 2.0

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

Problem #PRU-34875

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Geometry Plane geometry Polygons Hexagons Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Polygons Regular polygons

12–14 2.0

There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.