1983 IMO Problems
Problem 1
Find all functions defined on the set of positive reals which take positive real values and satisfy: for all ; and as .
Problem 2
Let be one of the two distinct points of intersection of two unequal coplanar circles and with centers and respectively. One of the common tangents to the circles touches at and at , while the other touches at and at . Let be the midpoint of and the midpoint of . Prove that .
Problem 3
Let and be positive integers, no two of which have a common divisor greater than . Show that is the largest integer which cannot be expressed in the form , where are non-negative integers.
Problem 4
Let be an equilateral triangle and the set of all points contained in the three segments , , and (including , , and ). Determine whether, for every partition of into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.
Problem 5
Is it possible to choose distinct positive integers, all less than or equal to , no three of which are consecutive terms of an arithmetic progression?
Problem 6
Let , and be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0. \] Determine when equality occurs.
See Also
1983 IMO (Problems) • Resources | ||
Preceded by 1982 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1984 IMO |
All IMO Problems and Solutions |