1985 IMO Problems
Problems of the 26th IMO Finland.
Contents
Day I
Problem 1
A circle has center on the side of the cyclic quadrilateral . The other three sides are tangent to the circle. Prove that .
Problem 2
Let and be given relatively prime natural numbers, . Each number in the set is colored either blue or white. It is given that
(i) for each , both and have the same color;
(ii) for each , both and have the same color.
Prove that all the numbers in have the same color.
Problem 3
For any polynomial with integer coefficients, the number of coefficients which are odd is denoted by . For , let . Prove that if are integers such that , then
.
Day II
Problem 4
Given a set of distinct positive integers, none of which has a prime divisor greater than , prove that contains a subset of elements whose product is the th power of an integer.
Problem 5
A circle with center passes through the vertices and of the triangle and intersects the segments and again at distinct points and respectively. Let be the point of intersection of the circumcircles of triangles and (apart from ). Prove that .
Problem 6
For every real number , construct the sequence by setting:
Prove that there exists exactly one value of which gives for all .
Resources
- 1985 IMO
- IMO 1985 problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1985 IMO (Problems) • Resources | ||
Preceded by 1984 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1986 IMO |
All IMO Problems and Solutions |