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…') |
|||
Line 47: | Line 47: | ||
[[1965 IMO Problems/Problem 6|Solution]] | [[1965 IMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | * [[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]] | ||
+ | |||
+ | {{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 |