Geometry Answers

Prove that at affine transformation parallel lines transform into parallel lines.

There are 400 points on a plane. Prove that different distances among them are not less then 15.

In every cell of a board with sizes 5x5 cells sits a bug. At some moment all bugs crawl into neighbouring (on horizontal or vertical) cells. Will an empty cell remain?

A chessboard is given. It is allowed simultaneously to paint into another colour all cells located in a square 2x2. Is it possible to get only one black cell on the board?

A chessboard is given. It is allowed simultaneously to paint into another colour cells of any vertical or horizontal lines. Is it possible to get a board where there is only one black cell?

There are three pucks on a plane A, B and C. Hockey player hits one of the pucks in such a way that it went between two other pucks and stopped at some point. Is it possible for pucks to return on its initial place after 25 hits?

Еhere are n unparallel lines in pairs on a plane. Prove that the angle between two of them is not more then (180°/n).

I need to know the arc length of m, ∠AOB=90(degrees) and R=6.

The program was written exactly as I requested. I will be requesting the same writer for my other programs if that is not a problem. The program was written in word document and compile with no additional adjustments or editing.

How do I determine that the line is perpendicular to the plane?