1968 IMO Problems
Problems of the 10th IMO 1968 in USSR.
Problem 1
Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.
Problem 2
Find all natural numbers such that the product of their digits (in decimal notation) is equal to .
Problem 3
Consider the system of equations with unknowns where are real and . Let . Prove that for this system
(a) if , there is no solution,
(b) if , there is exactly one solution,
(c) if , there is more than one solution.
Problem 4
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.
Problem 5
Let be a real-valued function defined for all real numbers such that, for some positive constant , the equation holds for all .
(a) Prove that the function is periodic (i.e., there exists a positive number such that for all ).
(b) For , give an example of a non-constant function with the required properties.
Problem 6
For every natural number , evaluate the sum (The symbol denotes the greatest integer not exceeding .)
- 1968 IMO
- IMO 1968 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1968 IMO (Problems) • Resources | ||
Preceded by 1967 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1969 IMO |
All IMO Problems and Solutions |