1967 IMO Problems
Problems of the 9th IMO 1967 in Yugoslavia.
Contents
Day I
Problem 1
Let be a parallelogram with side lengths , , and with . If is acute, prove that the four circles of radius with centers , , , cover the parallelogram if and only if
Problem 2
Prove that if one and only one edge of a tetrahedron is greater than , then its volume is .
Problem 3
Let , , be natural numbers such that is a prime greater than . Let . Prove that the product is divisible by the product .
Day II
Problem 4
Let and be any two acute-angled triangles. Consider all triangles that are similar to (so that vertices , , correspond to vertices , , , respectively) and circumscribed about triangle (where lies on , on , and on ). Of all such possible triangles, determine the one with maximum area, and construct it.
Problem 5
Consider the sequence , where in which , , , are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence are equal to zero. Find all natural numbers for which .
Problem 6
In a sports contest, there were medals awarded on successive days (). On the first day, one medal and of the remaining medals were awarded. On the second day, two medals and of the now remaining medals were awarded; and so on. On the -th and last day, the remaining medals were awarded. How many days did the contest last, and how many medals were awarded altogether?
- 1967 IMO
- IMO 1967 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1967 IMO (Problems) • Resources | ||
Preceded by 1966 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1968 IMO |
All IMO Problems and Solutions |