Difference between revisions of "1961 IMO Problems"
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==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
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| + | In the interior of [[triangle]] <math>ABC</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]] | ||
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===Problem 5=== | ===Problem 5=== | ||
Revision as of 11:33, 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
Problem 6
