# Difference between revisions of "1961 IMO Problems"

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===Problem 4=== | ===Problem 4=== | ||

− | In the interior of [[triangle]] <math> | + | In the interior of [[triangle]] <math>P_1 P_2 P_3</math> a [[point]] ''P'' is given. Let <math>Q_1,Q_2,Q_3</math> be the [[intersection]]s of <math>PP_1, PP_2,PP_3</math> with the opposing [[edge]]s of triangle <math>ABC</math>. Prove that among the [[ratio]]s <math>\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}</math> there exists one not larger than 2 and one not smaller than 2. |

[[1961 IMO Problems/Problem 4 | Solution]] | [[1961 IMO Problems/Problem 4 | Solution]] |

## Revision as of 18:27, 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 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 ?