Difference between revisions of "1965 IMO Problems"
5849206328x (talk | contribs) (Created page with '==Problem 1== Determine all values <math>x</math> in the interval <math>0\leq x\leq 2\pi </math> which satisfy the inequality <cmath>2\cos x \leq \left| \sqrt{1+\sin 2x} - \sqrt…') |
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[[1965 IMO Problems/Problem 6|Solution]] | [[1965 IMO Problems/Problem 6|Solution]] | ||
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+ | * [[1965 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1965 IMO 1965 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] | ||
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+ | {{IMO box|year=1965|before=[[1964 IMO]]|after=[[1966 IMO]]}} |
Latest revision as of 11:39, 29 January 2021
Problem 1
Determine all values in the interval which satisfy the inequality
Problem 2
Consider the system of equations with unknowns , , . The coefficients satisfy the conditions:
(a) , , are positive numbers;
(b) the remaining coefficients are negative numbers;
(c) in each equation, the sum of the coefficients is positive.
Prove that the given system has only the solution .
Problem 3
Given the tetrahedron whose edges and have lengths and respectively. The distance between the skew lines and is , and the angle between them is . Tetrahedron is divided into two solids by plane , parallel to lines and . The ratio of the distances of from and is equal to . Compute the ratio of the volumes of the two solids obtained.
Problem 4
Find all sets of four real numbers , , , such that the sum of any one and the product of the other three is equal to .
Problem 5
Consider with acute angle . Through a point perpendiculars are drawn to and , the feet of which are and respectively. The point of intersection of the altitudes of is . What is the locus of if is permitted to range over (a) the side , (b) the interior of ?
Problem 6
In a plane a set of points () is given. Each pair of points is connected by a segment. Let be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length . Prove that the number of diameters of the given set is at most .
- 1965 IMO
- IMO 1965 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1965 IMO (Problems) • Resources | ||
Preceded by 1964 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1966 IMO |
All IMO Problems and Solutions |