Let be distinct real numbers and be a real number. Given that three numbers among

coincide, prove that .

Two distinct positive integers are called "relatively consistent" if the larger one can be written as a sum of some distinct positive divisors of the other one. Show that there exist 2018 positive integers such that any two of them are "relatively consistent"

Let be the orthocenter of an acute angled triangle . Circumcircle of the triangle and the circle of diameter intersect at point , different from . Let be the midpoint of the small arc of the circumcircle of the triangle and let the midpoint of the large arc of the circumcircle of the triangle Prove that points are concyclic.

boxes are placed around a circle. At the first step we choose some boxes. At the second step for each chosen box we put a ball into the chosen box and into each of its two neighbouring boxes. Find the total number of possible distinct ball distributions which can be obtained in this way. (All balls are identical.)