1965 IMO Problems
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 |