There are 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?
Prove that there is no polyhedron that has exactly seven edges.
Prove that, with central symmetry, a circle transforms into a circle.
The opposite sides of a convex hexagon are pairwise equal and parallel. Prove that it has a centre of symmetry.
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.
Prove that a circle under the axial symmetry transforms into a circle.
A quadrilateral has an axis of symmetry. Prove that this quadrilateral is either an isosceles trapezoid or is symmetric with respect to its diagonal.
Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.
A passenger left his things in an automatic storage room, and when he came to get his things, it turned out that he had forgotten the code. He only remembers that in the code there were the numbers 23 and 37. To open the room, you need to correctly type a five-digit number. What is the least number of codes you need to sort through in order to open the room for sure?
Think of a way to finish constructing Pascal’s triangle upward.