# Difference between revisions of "1961 IMO Problems"

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===Problem 5=== | ===Problem 5=== | ||

− | + | Construct a triangle ''ABC'' if the following elements are given: <math>AC = b, AB = c</math>, and <math>\angle AMB = \omega \left(\omega < 90^{\circ}\right)</math> where ''M'' is the midpoint of ''BC''. Prove that the construction has a solution if and only if | |

+ | |||

+ | <math>b \tan{\frac{\omega}{2}} \le c < b</math> | ||

+ | In what case does equality hold? | ||

+ | [[1961 IMO Problems/Problem 5 | Solution]] | ||

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

## Revision as of 11:34, 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