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A circle cuts out equal chords on all sides of the quadrilateral. Prove that it is possible to inscribe a circle in that quadrilateral.
The radius of the inscribed circle of the triangle equals 1, heights are integers. Prove that the triangle is equilateral.
M is the middle of the side AB of the triangle ABC. Prove that CM = AB/2 only if the angle ∠ACB = 90°.
The distance between the centers of circles R1 and R2 is d. Prove that these circles are orthogonal only if d² = R1² + R2².
Isosceles trapezoids ABCD and A1B1C1D1 with respectively parallel sides are inscribed in a circle. Prove that AC = A1C1.
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?
