The International Olympiad has been held annually since 1959; the U.S. began participating in 1974, when the Sixteenth International Olympiad was held in Erfurt, G.D.R.In 1974 and 1975, the National Science Foundation funded a three week summer training session with Samuel L. Greizer of Rutgers University and Murray Klamkin of the University of Alberta as the U.S. teams' coaches. Summer training session in 1976, 1977 were funded by grants from the Army Research Office and the Office of naval Research. To date the U.S. teams have consistently placed among the top three national scores: second in 1974 (behind Hungary and the G.D.R.) and 1976 (behind the USSR and the G.D.R.) and 1976 (behind the USSR and Great Britain) and first in 1977.Members of the U.S. team are selected from the 100 top scorers on the Annual high School Examinations (see NML vols. 5, 17, 25) by subsequent competition in the U.S. Mathematical Olympiad.In this volume the demonstrably effective coach and prime mover in planning the participation of the U.S.A. in the I.M.O., Samuel L. Greitzer, has compiled all the IMO problems from the First through Nineteenth (1977) IMO and their solutions, some based on the contestants' papers.The problems are solvable by methods accessible to secondary school students in most nations, but insight and ingenuity are often required. A chronological examination of the questions throws some light on the changes and trends in secondary school mathematics curricula.