Difference between revisions of "1961 IMO Problems"
(→Problem 4) |
m (→Resources) |
||
| Line 58: | Line 58: | ||
[[1961 IMO Problems/Problem 6 | Solution]] | [[1961 IMO Problems/Problem 6 | Solution]] | ||
| − | == | + | == Resources == |
| + | * [[IMO Problems and Solutions, with authors]] | ||
| + | * [[Mathematics competition resources]] | ||
| + | |||
| + | {{IMO box|year=1961|before=[[1960 IMO]]|after=[[1962 IMO]]}} | ||
Revision as of 20:18, 20 August 2020
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 ?
Resources
| 1961 IMO (Problems) • Resources | ||
| Preceded by 1960 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1962 IMO |
| All IMO Problems and Solutions | ||
