1961 IMO Problems
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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 non-collinear 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 ?
