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
There are cities, airline companies in a country. Between any two cities, there is exactly one -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of
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
Find all pairs of integers satisfying: there exists an integer such that is divisible by for all positive integers
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