2011 IMO Problems
Problems of the 52st IMO 2011 in Amsterdam, Netherlands.
Given any set of four distinct positive integers, let be the sum of the positive integers in . Let denote the number of pairs with for which divides . Find all sets of four distinct positive integers which achieve the largest possible value of .
Author: Fernando Campos, Mexico
Let be a finite set of at least two points in the plane. Assume that no three points of are collinear. A windmill is a process that starts with a line going through a single point . The line rotates clockwise about the pivot until the first time that the line meets some other point belonging to . This point, , takes over as the new pivot, and the line now rotates clockwise about , until it next meets a point of . This process continues indefinitely. Show that we can choose a point in and a line going through such that the resulting windmill uses each point of as a pivot infinitely many times.
Author: Geoffrey Smith, United Kingdom
Let be a real-valued function defined on the set of real numbers that satisfies for all real numbers and . Prove that for all .
Author: Igor Voronovich, Belarus
Let be an integer. We are given a balance and weights of weight . We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done.
Author: Morteza Saghafian, Iran
Let be a function from the set of integers to the set of positive integers. Suppose that, for any two integers and , the difference is divisible by . Prove that, for all integers and with , the number is divisible by .
Author: Mahyar Sefidgaran, Iran
Let be an acute triangle with circumcircle . Let be a tangent line to , and let , and be the lines obtained by reflecting in the lines , and , respectively. Show that the circumcircle of the triangle determined by the lines , and is tangent to the circle .
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