Geometry Answers

There are two straight lines, which intersect. Can we draw a straight line which doesn’t belong to the plane, to which given lines belong, through the intersection point? Explain your answer.

Surface squares of two spheres refer as m:n. How do their volumes refer?

How many symmetry planes a regular (a) octahedron, (b) dodecahedron, (c) icosahedron has?

On the chessboard (8x8) all field centers are marked. Is it possible to break the board into the parts with thirteen straight lines so that within each of these parts there was no more than one marked point?

Five points are located within an equilateral triangle with the sides of 1. Prove that the distance between each pair of them is less than 0.5.

Nodes of the infinite checkered paper are painted in two colors. Prove that there are two horizontal and two vertical lines which intersect in the points of one color.

Is it possible to locate 1000 segments on the plane so that each segment at both ends can be rested strictly inside other segments?

N points and the middles of all segments with endpoints in them are marked on the plane. Prove that the amount of marked points is not less than (2n - 3).

On the plane there are several points, all the pairwise distance between them are different. Each of these points connect with the nearest. Can this set be a closed broken line?

N ≥ 3 points lie on a plane, but not all of them lie on a straight line. Prove that there is a circle passing through three of these points and containing none of the remaining points.