Find the number of integers in the set such that is a multiple of
Let be a sequence of nondecreasing positive integers such that for some positive integers and . Prove that there exists a positive integer such that
As shown in the figure below, the in-circle of is tangent to sides and at and respectively, and is the circumcenter of . Prove that .
There is a stone at each vertex of a given regular -gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the -gon
Circles and are tangent to each other externally at . Points and are on , lines and are tangent to at and , respectively, lines and meet at point
Let be non-negative real numbers. Prove that