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 17:27, 17 June 2016
Contents
[hide]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
?