1990 IMO Problems
Problems of the 1990 IMO.
Contents
Day I
Problem 1
Chords and of a circle intersect at a point inside the circle. Let be an interior point of the segment . The tangent line at to the circle through , , and intersects the lines and at and , respectively. If find in terms of .
Problem 2
Let and consider a set of distinct points on a circle. Suppose that exactly of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly points from . Find the smallest value of so that every such coloring of points of is good.
Problem 3
Determine all integers such that is an integer.
Day II
Problem 4
Let be the set of positive rational numbers. Construct a function such that for all , in .
Problem 5
Given an initial integer , two players, and , choose integers , , , alternately according to the following rules :
I.) Knowing , chooses any integer such that \[ n_{2k} \leq n_{2k + 1} \leq n_{2k}^2. \] II.) Knowing , chooses any integer such that \[ \frac {n_{2k + 1}}{n_{2k + 2}} \] is a prime raised to a positive integer power.
Player wins the game by choosing the number 1990; player wins by choosing the number 1. For which does :
a.) have a winning strategy?
b.) have a winning strategy?
c.) Neither player have a winning strategy?
Problem 6
Prove that there exists a convex 1990-gon with the following two properties :
a.) All angles are equal. b.) The lengths of the 1990 sides are the numbers , , , , in some order.
- 1990 IMO
- IMO 1990 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1990 IMO (Problems) • Resources | ||
Preceded by 1989 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1991 IMO |
All IMO Problems and Solutions |