1986 IMO Problems
Contents
Day 1
Problem 1
Let be any positive integer not equal to or . Show that one can find distinct in the set such that is not a perfect square.
Problem 2
Given a point in the plane of the triangle . Define for all . Construct a set of points such that is the image of under a rotation center through an angle clockwise for . Prove that if , then the triangle is equilateral.
Problem 3
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers respectively, and , then the following operation is allowed: are replaced by respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
Day 2
Problem 4
Let be adjacent vertices of a regular -gon () with center . A triangle , which is congruent to and initially coincides with , moves in the plane in such a way that and each trace out the whole boundary of the polygon, with remaining inside the polygon. Find the locus of .
Problem 5
Find all functions defined on the non-negative reals and taking non-negative real values such that: for , and for all .
Problem 6
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on is not greater than ?
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1986 IMO (Problems) • Resources | ||
Preceded by 1985 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1987 IMO |
All IMO Problems and Solutions |