Find the intersection of all rhombuses with vertices on a given ellipse . Coordinates of vertices of a parallelogram are integer, his area is equal to 3. Prove that they lies on one of diagonals of the parallelogram and divide this diagonal into three equal segments. Vertices of a triangle lies on a given ellipse, and the area of the triangle has the greatest possible value among triangles with this property. Prove that areas of three sectors complementary to the triangle inside the ellipse are equal.
Function is continuous on the segment and for all satisfies the inequality .
Prove that is constant. Find the least value for the integral where is the length of the projection for the radius-vector of the point to the tangent line at this point. A planet is received by rotation of the square around one of its diagonals. Prove that the functional equation admits no continuous on solutions . Prove that for each function continuous on .