Difference between revisions of "1961 IMO Problems"
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Revision as of 17:27, 17 June 2016
Contents
Day I
Problem 1
(Hungary) Solve the system of equations:
where and are constants. Give the conditions that and must satisfy so that (the solutions of the system) are distinct positive numbers.
Problem 2
Let a,b, and c be the lengths of a triangle whose area is S. Prove that
In what case does equality hold?
Problem 3
Solve the equation
where n is a given positive integer.
Day 2
Problem 4
In the interior of triangle a point P is given. Let be the intersections of with the opposing edges of triangle . Prove that among the ratios there exists one not larger than 2 and one not smaller than 2.
Problem 5
Construct a triangle ABC if the following elements are given: , and where M is the midpoint of BC. Prove that the construction has a solution if and only if
In what case does equality hold?
Problem 6
Consider a plane and three noncollinear points on the same side of ; suppose the plane determined by these three points is not parallel to . In plane take three arbitrary points . Let be the midpoints of segments ; Let be the centroid of the triangle . (We will not consider positions of the points such that the points do not form a triangle.) What is the locus of point as range independently over the plane ?