# Difference between revisions of "1961 IMO Problems"

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===Problem 6=== | ===Problem 6=== | ||

+ | Consider a plane <math>\epsilon</math> and three non-collinear points <math>A,B,C</math> on the same side of <math>\epsilon</math>; suppose the plane determined by these three points is not parallel to <math>\epsilon</math>. In plane <math>\epsilon</math> take three arbitrary points <math>A',B',C'</math>. Let <math>L,M,N</math> be the midpoints of segments <math>AA', BB', CC'</math>; Let <math>G</math> be the centroid of the triangle <math>LMN</math>. (We will not consider positions of the points <math>A', B', C'</math> such that the points <math>L,M,N</math> do not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>? | ||

[[1961 IMO Problems/Problem 6 | Solution]] | [[1961 IMO Problems/Problem 6 | Solution]] | ||

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==See Also== | ==See Also== |

## Revision as of 10:35, 12 October 2007

## 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 ?