1998 IMO Problems
Problems of the 1998 IMO.
Contents
[hide]Day I
Problem 1
In the convex quadrilateral , the diagonals and are perpendicular and the opposite sides and are not parallel. Suppose that the point , where the perpendicular bisectors of and meet, is inside . Prove that is a cyclic quadrilateral if and only if the triangles and have equal areas.
Problem 2
In a competition, there are contestants and judges, where is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose is a number such that, for any two judges, their ratings coincide for at most contestants. Prove that .
Problem 3
For any positive integer , let denote the number of positive divisors of (including 1 and itself). Determine all positive integers such that for some .
Day II
Problem 4
Determine all pairs of positive integers such that divides .
Problem 5
Let be the incenter of triangle . Let the incircle of touch the sides , , and at , , and , respectively. The line through parallel to meets the lines and at and , respectively. Prove that angle is acute.
Problem 6
Determine the least possible value of where is a function such that for all ,
See Also
1998 IMO (Problems) • Resources | ||
Preceded by 1997 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1999 IMO |
All IMO Problems and Solutions |