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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-57863

Problems Geometry Solid geometry Space transformations Isometry Reflectional symmetry

14–14 2.0

Prove that a circle under the axial symmetry transforms into a circle.

Problem #PRU-57864

Problems Geometry Solid geometry Space transformations Isometry Reflectional symmetry

14–14 2.0

A quadrilateral has an axis of symmetry. Prove that this quadrilateral is either an isosceles trapezoid or is symmetric with respect to its diagonal.

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.

Problem #PRU-87973

Problems Algebra and arithmetic Arithmetics. Mental maths Geometry Solid geometry Volume Ratio of volumes Space transformations Similarity

10–12 2.0

When Gulliver came to Lilliput, he found that everything was exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into the matchbox of Gulliver?