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 10:34, 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
