1988 IMO Problems
Problems of the 1988 IMO.
Contents
Day I
Problem 1
Consider 2 concentric circle radii and () with centre Fix on the small circle and consider the variable chord of the small circle. Points and lie on the large circle; are collinear and is perpendicular to
i.) For which values of is the sum extremal?
ii.) What are the possible positions of the midpoints of and of as varies?
Problem 2
Let be an even positive integer. Let be sets having elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly zeros?
Problem 3
A function defined on the positive integers (and taking positive integers values) is given by:
for all positive integers Determine with proof the number of positive integers for which
Day II
Problem 4
Show that the solution set of the inequality \[ \sum^{70}_{k = 1} \frac {k}{x - k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.
Problem 5
In a right-angled triangle let be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles intersect the sides at the points respectively. If and dnote the areas of triangles and respectively, show that
Problem 6
Let and be two positive integers such that divides . Show that is a perfect square.
- 1988 IMO
- IMO 1988 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1988 IMO (Problems) • Resources | ||
Preceded by 1987 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1989 IMO |
All IMO Problems and Solutions |