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Found: 3

Problem #PRU-35643

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Plane geometry Polygons Regular polygons

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

Problem #PRU-79399

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Plane geometry Polygons Regular polygons Quadrilateral Trapeziums

In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.

Problem #PRU-98408

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Diameter, chords, and secants Chords and secants (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Regular polygons

In a regular shape with 25 vertices, all the diagonals are drawn.

Prove that there are no nine diagonals passing through one interior point of the shape.