Difference between revisions of "1985 IMO Problems"
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* [[1985 IMO]] | * [[1985 IMO]] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1985 IMO 1985 problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1985 IMO 1985 problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1985|before=[[1984 IMO]]|after=[[1986 IMO]]}} |
Revision as of 22:54, 29 January 2021
Problems of the 26th IMO Finland.
Contents
[hide]Day I
Problem 1
A circle has center on the side of the cyclic quadrilateral
. The other three sides are tangent to the circle. Prove that
.
= Problem 2
Let and
be given relatively prime natural numbers,
. Each number in the set
is colored either blue or white. It is given that
(i) for each , both
and
have the same color;
(ii) for each , both
and
have the same color.
Prove that all the numbers in have the same color.
Problem 3
For any polynomial with integer coefficients, the number of coefficients which are odd is denoted by
. For
, let
. Prove that if
are integers such that
, then
.
Day II
Problem 4
Given a set of
distinct positive integers, none of which has a prime divisor greater than
, prove that
contains a subset of
elements whose product is the
th power of an integer.
Problem 5
A circle with center passes through the vertices
and
of the triangle
and intersects the segments
and
again at distinct points
and
respectively. Let
be the point of intersection of the circumcircles of triangles
and
(apart from
). Prove that
.
Problem 6
For every real number , construct the sequence
by setting:
Prove that there exists exactly one value of which gives
for all
.
Resources
- 1985 IMO
- IMO 1985 problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1985 IMO (Problems) • Resources | ||
Preceded by 1984 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1986 IMO |
All IMO Problems and Solutions |