1969 IMO Problems
Problems of the 11th IMO 1969 in Romania.
Problem 1
Prove that there are infinitely many natural numbers with the following property: the number is not prime for any natural number .
Problem 2
Let be real constants, a real variable, and Given that , prove that for some integer .
Problem 3
For each value of , find necessary and sufficient conditions on the number so that there exists a tetrahedron with k edges of length , and the remaining edges of length 1.
Problem 4
A semicircular arc is drawn on as diameter. is a point on other than and , and is the foot of the perpendicular from to . We consider three circles, , all tangent to the line . Of these, is inscribed in , while and are both tangent to and to , one on each side of . Prove that , and have a second tangent in common.
Problem 5
Given points in the plane such that no three are collinear. Prove that there are at least convex quadrilaterals whose vertices are four of the given points.
Problem 6
Prove that for all real numbers , with , the inequality is satisfied. Give necessary and sufficient conditions for equality.
- 1969 IMO
- IMO 1969 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1969 IMO (Problems) • Resources | ||
Preceded by 1968 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1970 IMO |
All IMO Problems and Solutions |