Difference between revisions of "1975 IMO Problems"
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[[1975 IMO Problems/Problem 6|Solution]] | [[1975 IMO Problems/Problem 6|Solution]] | ||
− | * [[1975 IMO]] * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1975 IMO 1975 Problems on the Resources page] * [[IMO Problems and Solutions, with authors]] * [[Mathematics competition resources]] {{IMO box|year=1975|before=[[1974 IMO]]|after=[[1976 IMO]]}} | + | * [[1975 IMO]] |
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1975 IMO 1975 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1975|before=[[1974 IMO]]|after=[[1976 IMO]]}} |
Latest revision as of 16:06, 29 January 2021
Problems of the 17th IMO 1975 in Bulgaria.
Problem 1
Let be real numbers such that
Prove that, if
is any permutation of
, then
Problem 2
Let be an infinite increasing sequence of positive integers. Prove that for every
there are infinitely many
which can be written in the form
with
positive integers and
.
Problem 3
On the sides of an arbitrary triangle , triangles
are constructed externally with
. Prove that
and
.
Problem 4
When is written in decimal notation, the sum of its digits is
. Let
be the sum of the digits of
. Find the sum of the digits of
. (
and
are written in decimal notation.)
Problem 5
Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.
Problem 6
Find all polynomials , in two variables, with the following properties:
(i) for a positive integer and all real
(that is,
is homogeneous of degree
),
(ii) for all real
(iii)
- 1975 IMO
- IMO 1975 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1975 IMO (Problems) • Resources | ||
Preceded by 1974 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1976 IMO |
All IMO Problems and Solutions |