1982 IMO Problems
Contents
Day 1
Problem 1
The function is defined on the positive integers and takes non-negative integer values. and for all Determine .
Problem 2
A non-isosceles triangle has sides , , with the side lying opposite to the vertex . Let be the midpoint of the side , and let be the point where the inscribed circle of triangle touches the side . Denote by the reflection of the point in the interior angle bisector of the angle . Prove that the lines , and are concurrent. Solution
Problem 3
Consider infinite sequences of positive reals such that and .
a) Prove that for every such sequence there is an such that:
b) Find such a sequence such that for all : Solution
Day 2
Problem 4
Prove that if is a positive integer such that the equation has a solution in integers , then it has at least three such solutions. Show that the equation has no solutions in integers for . Solution
Problem 5
The diagonals and of the regular hexagon are divided by inner points and respectively, so that Determine if and are collinear. Solution
Problem 6
Let be a square with sides length . Let be a path within which does not meet itself and which is composed of line segments with . Suppose that for every point on the boundary of there is a point of at a distance from no greater than . Prove that there are two points and of such that the distance between and is not greater than and the length of the part of which lies between and is not smaller than . Solution
See also
1982 IMO (Problems) • Resources | ||
Preceded by 1981 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1983 IMO Problems |
All IMO Problems and Solutions |