1 The IMO Compendium - IMOmath

1 1 Djuki Jankovi Mati Petrovi Problem Books in MathematicsDu an Djuki Vladimir Jankovi Ivan Mati Nikola Petrovi The IMO Compendium A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009, Second EditionProblem Books in MathematicsThe IMO CompendiumDu an Djuki Vladimir Jankovi Ivan Mati Nikola Petrovi Mathematics ABCDPBMThe IMO Compendium A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009 Second Edition! e IMO Compendium is the ultimate collection of challenging high school level mathemat-ics problems. It is an invaluable resource, not only for students preparing for competitions, but for anyone who loves and appreciates math. Training for mathematical olympiads is enjoyed by talented students throughout the world. Olympiads have become one of the primary ways to recognize and develop talented youth with a potential to excel in areas that require abstract thinking.

2 Although the problems appearing at IMO do not involve advanced mathematics, they must be di! cult and their solutions must arise from creative and clever insights rather than tedious preparation for the distinguished International Mathematical Olympiad (IMO) competi-tion, each participating country selects the top six high school students every year through a series of national olympiads. " ese students are invited to participate in the IMO, usually held in July. " e IMO is a two-day contest where each day competitors are given three problems which they work on independently. " e IMO host country appoints a special committee to which each country submits up to six problems. From this composite longlist of prob-lems, a shortlist of #$-%& problems is created. " e jury, consisting of one professor from each country, makes the ' nal selection from the shortlist a few days before the IMO begins.

3 " e IMO has sparked a burst of creativity among enthusiasts to create new and interest-ing mathematics problems. It can be safely said that the IMO and shortlisted problems are among the well-cra( ed problems created in a given year. " is book attempts to gather all of these problems with their solutions. In addition, the book contains all the available longlist problems, for a total of more than #&&& the reviews of the ' rst edition:"" e International Mathematical Olympiad, or IMO is the premier international competi-tion for talented high school mathematics students.. " is book collects statements and solutions of all of the problems ever set in the IMO, together with many problems proposed for the contest.. serves as a vast repository of problems at the Olympiad level, useful both to students .. and to faculty looking for hard elementary problems.)

4 No library will want to be without a copy, nor will anyone involved in mathematics competitions .." (Fernando Q. Gouv a, MathDL, March, #&&))2nd The International Mathematical Olympiad .. The IMO Compendium .. Algebra .. Polynomials .. Recurrence Relations .. Inequalities .. Groups and Fields.. Analysis .. Geometry .. Triangle Geometry .. Vectors in Geometry.. Barycenters .. Quadrilaterals .. Circle Geometry .. Inversion .. Geometric Inequalities .. Trigonometry .. Formulas in Geometry .. Number Theory .. Divisibility and Congruences .. Exponential Congruences .. Quadratic Diophantine Equations .. Farey Sequences .. Combinatorics .. Counting of Objects .. Graph Theory .. IMO 1959 .. Contest Problems .. IMO 1960.

5 Contest Problems .. IMO 1961 .. Contest Problems .. IMO 1962 .. Contest Problems .. IMO 1963 .. Contest Problems .. IMO 1964 .. Contest Problems .. IMO 1965 .. Contest Problems .. IMO 1966 .. Contest Problems .. Some Longlisted Problems 1959 1966 .. IMO 1967 .. Contest Problems .. Longlisted Problems .. IMO 1968 .. Contest Problems .. Shortlisted Problems .. IMO 1969 .. Contest Problems .. Longlisted Problems .. IMO 1970 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1971 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1972 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1973 .. Contest Problems .. Shortlisted Problems .. IMO 1974.

6 Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1975 .. Contest Problems .. Shortlisted Problems .. IMO 1976 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1977 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1978 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1979 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1981 .. Contest Problems .. Shortlisted Problems .. IMO 1982 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1983 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1984 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1985 .. Contest Problems.

7 Longlisted Problems .. Shortlisted Problems .. IMO 1986 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1987 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1988 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1989 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1990 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1991 .. Contest Problems .. Shortlisted Problems .. IMO 1992 .. Contest Problems .. Longlisted Problems .. Shortlisted Problems .. IMO 1993 .. Contest Problems .. Shortlisted Problems .. IMO 1994 .. Contest Problems .. Shortlisted Problems .. IMO 1995 .. Contest Problems .. Shortlisted Problems .. IMO 1996 .. Contest Problems.

8 Shortlisted Problems .. IMO 1997 .. Contest Problems .. Shortlisted Problems .. IMO 1998 .. Contest Problems .. Shortlisted Problems .. IMO 1999 .. Contest Problems .. Shortlisted Problems .. IMO 2000 .. Contest Problems .. Shortlisted Problems .. 300 Contents IMO 2001 .. Contest Problems .. Shortlisted Problems .. IMO 2002 .. Contest Problems .. Shortlisted Problems .. IMO 2003 .. Contest Problems .. Shortlisted Problems .. IMO 2004 .. Contest Problems .. Shortlisted Problems .. IMO 2005 .. Contest Problems .. Shortlisted Problems .. IMO 2006 .. Contest Problems .. Shortlisted Problems .. IMO 2007 .. Contest Problems .. Shortlisted Problems .. IMO 2008 .. Contest Problems .. Shortlisted Problems .. IMO 2009 .. Contest Problems .. Shortlisted Problems.

9 Contest Problems 1959 .. Contest Problems 1960 .. Contest Problems 1961 .. Contest Problems 1962 .. Contest Problems 1963 .. Contest Problems 1964 .. Contest Problems 1965 .. Contest Problems 1966 .. Longlisted Problems 1967 .. Shortlisted Problems 1968 .. Contest Problems 1969 .. Shortlisted Problems 1970 .. Shortlisted Problems 1971 .. Shortlisted Problems 1972 .. Shortlisted Problems 1973 .. Shortlisted Problems 1974 .. 407 XIV Shortlisted Problems 1975 .. Shortlisted Problems 1976 .. Longlisted Problems 1977 .. Shortlisted Problems 1978 .. Shortlisted Problems 1979 .. Shortlisted Problems 1981 .. Shortlisted Problems 1982 .. Shortlisted Problems 1983 .. Shortlisted Problems 1984 .. Shortlisted Problems 1985 .. Shortlisted Problems 1986 .. Shortlisted Problems 1987.

10 Shortlisted Problems 1988 .. Shortlisted Problems 1989 .. Shortlisted Problems 1990 .. Shortlisted Problems 1991 .. Shortlisted Problems 1992 .. Shortlisted Problems 1993 .. Shortlisted Problems 1994 .. Shortlisted Problems 1995 .. Shortlisted Problems 1996 .. Shortlisted Problems 1997 .. Shortlisted Problems 1998 .. Shortlisted Problems 1999 .. Shortlisted Problems 2000 .. Shortlisted Problems 2001 .. Shortlisted Problems 2002 .. Shortlisted Problems 2003 .. Shortlisted Problems 2004 .. Shortlisted Problems 2005 .. Shortlisted Problems 2006 .. Shortlisted Problems 2007 .. Shortlisted Problems 2008 .. Shortlisted Problems 2009 .. Notation .. Abbreviations.. 8053223 The Forty-Sixth IMOM rida, Mexico, July 8 19, Contest ProblemsFirst Day (July 13)1. Six points are chosen on the sides of an equilateral triangleABC:A1,A2onBC;B1,B2onCA;C1, equal side lengths.