International Mathematical Olympiad 2011
Problems
Problems
Problem 1: Given any set of four distinct positive integers, we denote the sum by . 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 .
Proposed by Fernando Campos, from Mexico
Proposed by Fernando Campos, from Mexico
Problem 2: 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.
Proposed by Geoffrey Smith, United Kingdom
Proposed by Geoffrey Smith, United Kingdom
Problem 3: Let be a real-valued function defined on the set of real numbers that satisfies
for all real numbers and . Prove that for all .
Proposed by Igor Voronovich, Belarus
for all real numbers and . Prove that for all .
Proposed by Igor Voronovich, Belarus
Problem 4: 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.
Proposed by Morteza Saghafian, Iran
Determine the number of ways in which this can be done.
Proposed by Morteza Saghafian, Iran
Problem 5: 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 .
Proposed by Mahyar Sefidgaran, Iran
Proposed by Mahyar Sefidgaran, Iran
Download Now: IMO 2011 Problems and Solutions
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