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Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Discrete Mathematics Combinatorics Systems of points and line segments Systems of points and line segments (counterexamples)

13–13 2.0

a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?

b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?

Problem #PRU-35339

Problems Geometry Solid geometry Space transformations Isometry Isometry helps solve the problem Methods Pigeonhole principle Pigeonhole principle (area and volume) Point symmetry

13–15 2.0

On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.

Problem #PRU-57833

Problems Geometry Solid geometry Space transformations Isometry Reflectional symmetry

12–14 2.0

Prove that, with central symmetry, a circle transforms into a circle.

Problem #PRU-57835

Problems Geometry Solid geometry Space transformations Isometry Point symmetry Reflectional symmetry

12–14 2.0

The opposite sides of a convex hexagon are pairwise equal and parallel. Prove that it has a centre of symmetry.

Problem #PRU-57836

Problems Geometry Solid geometry Space transformations Isometry Point symmetry

13–15 3.0

A parallelogram \(ABCD\) and a point \(E\) are given. Through the points \(A, B, C, D\), lines parallel to the straight lines \(EC, ED, EA,EB\), respectively, are drawn. Prove that they intersect at one point.

Problem #PRU-57866

Problems Geometry Solid geometry Space transformations Isometry Point symmetry Reflectional symmetry

13–15 2.0

Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.