Transcription
ABCDPBMDušan Djukić · Vladimir Janković · Ivan Matić · Nikola PetrovićThe IMO CompendiumA Collection of Problems Suggested for The InternationalMathematical Olympiads: 1959-2009, Second EditionThe IMO Compendium is the ultimate collection of challenging high school level mathematics 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 isenjoyed by talented students throughout the world. Olympiads have become one of theprimary ways to recognize and develop talented youth with a potential to excel in areasthat require abstract thinking. Although the problems appearing at IMO do not involveadvanced mathematics, they must be difficult and their solutions must arise from creativeand clever insights rather than tedious calculations.The IMO has sparked a burst of creativity among enthusiasts to create new and interesting mathematics problems. It can be safely said that the IMO and shortlisted problems areamong the well-crafted problems created in a given year. This book attempts to gather all ofthese problems with their solutions. In addition, the book contains all the available longlistproblems, for a total of more than 2000 problems.From the reviews of the first edition:"The International Mathematical Olympiad, or IMO is the premier international competition for talented high school mathematics students. This book collects statements andsolutions of all of the problems ever set in the IMO, together with many problems proposedfor the contest. serves as a vast repository of problems at the Olympiad level, useful bothto students and to faculty looking for hard elementary problems. No library will want tobe without a copy, nor will anyone involved in mathematics competitions "— (Fernando Q. Gouvêa, MathDL, March, 2006)1The IMO CompendiumIn preparation for the distinguished International Mathematical Olympiad (IMO) competition, each participating country selects the top six high school students every year through aseries of national olympiads. These students are invited to participate in the IMO, usually heldin July. The IMO is a two-day contest where each day competitors are given three problemswhich they work on independently. The IMO host country appoints a special committeeto which each country submits up to six problems. From this composite “longlist” of problems, a “shortlist” of 25-30 problems is created. The jury, consisting of one professor fromeach country, makes the final selection from the shortlist a few days before the IMO begins.Djukić · JankovićMatić · PetrovićProblem Books in MathematicsMathematicsISBN 978-1-4419-9853-89 781441 9985382nd Ed.Problem Books in MathematicsDušan DjukićVladimir JankovićIvan MatićNikola PetrovićThe IMOCompendiumA Collection of Problems Suggestedfor The International MathematicalOlympiads: 1959-2009Second Edition
Contents1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 The International Mathematical Olympiad . . . . . . . . . . . . . . . . . . . . . .1.2 The IMO Compendium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1122Basic Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.2 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.4 Groups and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.1 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.2 Vectors in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.3 Barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.4 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.5 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.7 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.8 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.9 Formulas in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.1 Divisibility and Congruences . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.2 Exponential Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.3 Quadratic Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . .2.4.4 Farey Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5.1 Counting of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55567910121213141415161717181919202122222223
X3ContentsProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 IMO 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 IMO 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3 IMO 1961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4 IMO 1962 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5 IMO 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6 IMO 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7 IMO 1965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.8 IMO 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.8.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.8.2 Some Longlisted Problems 1959–1966 . . . . . . . . . . . . . . . . . .3.9 IMO 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.9.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.9.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.10 IMO 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.10.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.10.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.11 IMO 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.11.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.11.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.12 IMO 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.12.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.12.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.12.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.13 IMO 1971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.13.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.13.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.13.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.14 IMO 1972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.14.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.14.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.14.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.15 IMO 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.15.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.15.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.16 IMO 1974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.16.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.16.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35353616162687070717679797983868687898990
Contents3.16.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.17 IMO 1975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.17.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.17.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.18 IMO 1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.18.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.18.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.18.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.19 IMO 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.19.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.19.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.19.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.20 IMO 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.20.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.20.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.20.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.21 IMO 1979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.21.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.21.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.21.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.22 IMO 1981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.22.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.22.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.23 IMO 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.23.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.23.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.23.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.24 IMO 1983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.24.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.24.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.24.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.25 IMO 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.25.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.25.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.25.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.26 IMO 1985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.26.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.26.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.26.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.27 IMO 1986 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.27.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.27.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.27.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.28 IMO 1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.28.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8165168168168177181181182188192192
83.393.403.413.28.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.28.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.29.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.29.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.29.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.30.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.30.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.30.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.31.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.31.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.31.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.32.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.32.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.33.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.33.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.33.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.34.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.34.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.35.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.35.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.36.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.36.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.37.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.37.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.38.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.38.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.39.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.39.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.40.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.40.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .IMO 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.41.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.41.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286286287291291291295295295300300300
Contents4XIII3.42 IMO 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.42.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.42.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.43 IMO 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.43.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.43.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.44 IMO 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.44.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.44.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.45 IMO 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.45.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.45.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.46 IMO 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.46.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.46.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.47 IMO 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.47.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.47.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.48 IMO 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.48.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.48.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.49 IMO 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.49.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.49.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.50 IMO 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.50.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.50.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26326331331332336336337341341341Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1 Contest Problems 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2 Contest Problems 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3 Contest Problems 1961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.4 Contest Problems 1962 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5 Contest Problems 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.6 Contest Problems 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7 Contest Problems 1965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.8 Contest Problems 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.9 Longlisted Problems 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.10 Shortlisted Problems 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.11 Contest Problems 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.12 Shortlisted Problems 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.13 Shortlisted Problems 1971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.14 Shortlisted Problems 1972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.15 Shortlisted Problems 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.16 Shortlisted Problems 1974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07
rtlisted Problems 1975Shortlisted Problems 1976Longlisted Problems 1977Shortlisted Problems 1978Shortlisted Problems 1979Shortlisted Problems 1981Shortlisted Problems 1982Shortlisted Problems 1983Shortlisted Problems 1984Shortlisted Problems 1985Shortlisted Problems 1986Shortlisted Problems 1987Shortlisted Problems 1988Shortlisted Problems 1989Shortlisted Problems 1990Shortlisted Problems 1991Shortlisted Problems 1992Shortlisted Problems 1993Shortlisted Problems 1994Shortlisted Problems 1995Shortlisted Problems 1996Shortlisted Problems 1997Shortlisted Problems 1998Shortlisted Problems 1999Shortlisted Problems 2000Shortlisted Problems 2001Shortlisted Problems 2002Shortlisted Problems 2003Shortlisted Problems 2004Shortlisted Problems 2005Shortlisted Problems 2006Shortlisted Problems 2007Shortlisted Problems 2008Shortlisted Problems 4765777ANotation and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791A.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792BCodes of the Countries of Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795CAuthors of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
3223 Problems3.46 The Forty-Sixth IMOMérida, Mexico, July 8–19, 20053.46.1 Contest ProblemsFirst Day (July 13)1. Six points are chosen on the sides of an equilateral triangle ABC: A1 , A2 on BC;B1 , B2 on CA; C1 ,C2 on AB. These points are vertices of a convex hexagonA1 A2 B1 B2C1C2 with equal side lengths. Prove that the lines A1 B2 , B1C2 andC1 A2 are concurrent.2. Let a1 , a2 , . . . be a sequence of integers with infinitely many positive terms andinfinitely many negative terms. Suppose that for each positive integer n, the numbers a1 , a2 , . . . , an leave n different remainders on division by n. Prove that eachinteger occurs exactly once in the sequence.3. Let x, y, and z be positive real numbers such that xyz 1. Prove thatx5 x2y5 y2z5 z2 0.x5 y2 z2 y5 z2 x2 z5 x2 y2Second Day (July 14)4. Consider the sequence a1 , a2 , . . . defined byan 2n 3n 6n 1 (n 1, 2, . . . ).Determine all positive integers that are relatively prime to every term of thesequence.5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in lengthand not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE DF. The lines AC and BD meet at P; the lines BD and EFmeet at Q; the lines EF and AC meet at R. Consider all the triangles PQR as Eand F vary. Show that the circumcircles of these triangles have a common pointother than P.6. In a mathematical competition, six problems were posed to the contestants. Eachpair of problems was solved by more than 2/5 of the contestants. Nobody solvedall six problems. Show that there were at least two contestants who each solvedexactly five problems.3.46.2 Shortlisted Problems1. A1 (ROU) Find all monic polynomials p(x) with integer coefficients of degreetwo for which there exists a polynomial q(x) with integer coefficients such thatp(x)q(x) is a polynomial having all coefficients 1.
3.46 IMO 20053232. A2 (BGR) Let R denote the set of positive real numbers. Determine all functions f : R R such thatf (x) f (y) 2 f (x y f (x))for all positive real numbers x and y.3. A3 (CZE) Four real numbers p, q, r, s satisfyp q r s 9 andp2 q2 r2 s2 21.Prove that ab cd 2 holds for some permutation (a, b, c, d) of (p, q, r, s).4. A4 (IND) Find all functions f : R R satisfying the equationf (x y) f (x) f (y) f (xy) 2xy 1for all real x and y.5. A5 (KOR)IMO3 Let x, y and z be positive real numbers such that xyz 1. Provethaty5 y2z5 z2x5 x2 0.x5 y2 z2 y5 z2 x2 z5 x2 y26. C1 (AUS) A house has an even number of lamps distributed among its roomsin such a way that there are at least three lamps in every room. Each lamp sharesa switch with exactly one other lamp, not necessarily from the same room. Eachchange in the switch shared by two lamps changes their states simultaneously.Prove that for every initial state of the lamps there exists a sequence of changesin some of the switches at the end of which each room contains lamps
"" 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
