Difference between revisions of "1973 IMO Problems"
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+ | * [[1973 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1973 IMO 1973 Problems on the Resources page] * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1973|before=[[1972 IMO]]|after=[[1974 IMO]]}} |
Revision as of 14:43, 29 January 2021
Problems of the 15th IMO 1973 in USSR.
Problem 1
Point lies on line
;
are unit vectors such that points
all lie in a plane containing
and on one side of
. Prove that if
is odd,
Here
denotes the length of vector
.
Problem 2
Determine whether or not there exists a finite set of points in space not lying in the same plane such that, for any two points
and
of
, one can select two other points
and
of
so that lines
and
are parallel and not coincident.
Problem 3
Let and
be real numbers for which the equation
has at least one real solution. For all such pairs
, find the minimum value of
.
Problem 4
A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission?
Problem 5
is a set of non-constant functions of the real variable
of the form
and
has the following properties:
(a) If and
are in
, then
is in
; here
.
(b) If is in
, then its inverse
is in
; here the inverse of
is
.
(c) For every in
, there exists a real number
such that
.
Prove that there exists a real number such that
for all
in
.
Problem 6
Let be
positive numbers, and let
be a given real number such that
. Find
numbers
for which
(a) for
,
(b) for
,
(c) .
- 1973 IMO
- IMO 1973 Problems on the Resources page * IMO Problems and Solutions, with authors
- Mathematics competition resources
1973 IMO (Problems) • Resources | ||
Preceded by 1972 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1974 IMO |
All IMO Problems and Solutions |