Difference between revisions of "1985 IMO Problems"
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=== Problem 2 === | === Problem 2 === | ||
| − | Let <math>n</math> and <math>k</math> be given relatively prime natural numbers, <math>n | + | Let <math>n</math> and <math>k</math> be given relatively prime natural numbers, <math>k < n</math>. Each number in the set <math>M = \{ 1,2, \ldots , n-1 \} </math> is colored either blue or white. It is given that |
(i) for each <math> i \in M </math>, both <math>i </math> and <math>n-i </math> have the same color; | (i) for each <math> i \in M </math>, both <math>i </math> and <math>n-i </math> have the same color; | ||
Revision as of 14:13, 9 January 2015
Problems of the 26th IMO Finland.
Contents
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 .
