Difference between revisions of "1961 IMO Problems"
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===Problem 3=== | ===Problem 3=== | ||
+ | Solve the equation | ||
+ | |||
+ | <math>\cos^n{x} - \sin^n{x} = 1</math> | ||
+ | |||
+ | where ''n'' is a given positive integer. | ||
[[1961 IMO Problems/Problem 3 | Solution]] | [[1961 IMO Problems/Problem 3 | Solution]] | ||
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==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
+ | |||
+ | 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]] | ||
+ | ===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]] | [[1961 IMO Problems/Problem 5 | Solution]] | ||
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− | |||
===Problem 6=== | ===Problem 6=== | ||
+ | Consider a plane <math>\epsilon</math> and three non-collinear points <math>A,B,C</math> on the same side of <math>\epsilon</math>; suppose the plane determined by these three points is not parallel to <math>\epsilon</math>. In plane <math>\epsilon</math> take three arbitrary points <math>A',B',C'</math>. Let <math>L,M,N</math> be the midpoints of segments <math>AA', BB', CC'</math>; Let <math>G</math> be the centroid of the triangle <math>LMN</math>. (We will not consider positions of the points <math>A', B', C'</math> such that the points <math>L,M,N</math> do not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>? | ||
[[1961 IMO Problems/Problem 6 | Solution]] | [[1961 IMO Problems/Problem 6 | Solution]] | ||
+ | == Resources == | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] | ||
− | + | {{IMO box|year=1961|before=[[1960 IMO]]|after=[[1962 IMO]]}} | |
− | == |
Latest revision as of 21:18, 20 August 2020
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 ?
Resources
1961 IMO (Problems) • Resources | ||
Preceded by 1960 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1962 IMO |
All IMO Problems and Solutions |