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Difference between revisions of "1985 IMO Problems"

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=== Problem 1 ===
 
=== Problem 1 ===
  
−
A circle has center on the side <math>\displaystyle AB</math> of the cyclic quadrilateral <math>\displaystyle ABCD</math>.  The other three sides are tangent to the circle.  Prove that <math>\displaystyle AD + BC = AB</math>.
+
A circle has center on the side <math>AB</math> of the cyclic quadrilateral <math>ABCD</math>.  The other three sides are tangent to the circle.  Prove that <math>AD + BC = AB</math>.
  
 
[[1985 IMO Problems/Problem 1 | Solution]]
 
[[1985 IMO Problems/Problem 1 | Solution]]
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=== Problem 2 ===
 
=== Problem 2 ===
  
−
Let <math>\displaystyle n</math> and <math>\displaystyle k</math> be given relatively prime natural numbers, <math>\displaystyle n < k</math>.  Each number in the set <math>\displaystyle M = \{ 1,2, \ldots , n-1 \} </math> is colored either blue or white.  It is given that
+
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> \displaystyle i </math> and <math> \displaystyle 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;
  
−
(ii) for each <math> i \in M, i \neq k </math>, both <math> \displaystyle i </math> and <math> \displaystyle |i-j| </math> have the same color.
+
(ii) for each <math> i \in M, i \neq k </math>, both <math>i </math> and <math>|i-k| </math> have the same color.
  
−
Prove that all number in <math>\displaystyle M</math> have the same color.
+
Prove that all the numbers in <math>M</math> have the same color.
  
 
[[1985 IMO Problems/Problem 2 | Solution]]
 
[[1985 IMO Problems/Problem 2 | Solution]]
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=== Problem 3 ===
 
=== Problem 3 ===
  
−
For any polynomial <math> P(x) = a_0 + a_1 x + \cdots + a_k x^k </math> with integer coefficients, the number of coefficients which are odd is denoted by <math> \displaystyle w(P) </math>.  For <math> i = 0, 1, \ldots </math>, let <math> \displaystyle Q_i (x) = (1+x)^i </math>.  Prove that if <math> i_1, i_2, \ldots , i_n </math> are integers such that <math> 0 \leq i_1 < i_2 < \cdots < i_n </math>, then
+
For any polynomial <math> P(x) = a_0 + a_1 x + \cdots + a_k x^k </math> with integer coefficients, the number of coefficients which are odd is denoted by <math>w(P) </math>.  For <math> i = 0, 1, \ldots </math>, let <math>Q_i (x) = (1+x)^i </math>.  Prove that if <math> i_1, i_2, \ldots , i_n </math> are integers such that <math> 0 \leq i_1 < i_2 < \cdots < i_n </math>, then
  
 
<center>
 
<center>
 
<math>
 
<math>
−
\displaystyle w(Q_{i_1} + Q_{i_2} + \cdots + Q_{i_n}) \ge w(Q_{i_1})
+
w(Q_{i_1} + Q_{i_2} + \cdots + Q_{i_n}) \ge w(Q_{i_1})
 
</math>.
 
</math>.
 
</center>
 
</center>
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=== Problem 4 ===
 
=== Problem 4 ===
 +
 +
Given a set <math>M</math> of <math>1985</math> distinct positive integers, none of which has a prime divisor greater than <math>23</math>, prove that <math>M</math> contains a subset of <math>4</math> elements whose product is the <math>4</math>th power of an integer.
  
 
[[1985 IMO Problems/Problem 4 | Solution]]
 
[[1985 IMO Problems/Problem 4 | Solution]]
  
 
=== Problem 5 ===
 
=== Problem 5 ===
 +
 +
A circle with center <math>O</math> passes through the vertices <math>A</math> and <math>C</math> of the triangle <math>ABC</math> and intersects the segments <math>AB</math> and <math>BC</math> again at distinct points <math>K</math> and <math>N</math> respectively. Let <math>M</math> be the point of intersection of the circumcircles of triangles <math>ABC</math> and <math>KBN</math> (apart from <math>B</math>). Prove that <math>\angle OMB = 90^{\circ}</math>.
  
 
[[1985 IMO Problems/Problem 5 | Solution]]
 
[[1985 IMO Problems/Problem 5 | Solution]]
  
 
=== Problem 6 ===
 
=== Problem 6 ===
 +
 +
For every real number <math>x_1</math>, construct the sequence <math>x_1,x_2,\ldots</math> by setting:
 +
 +
<center>
 +
<math>
 +
x_{n + 1} = x_n(x_n + {1\over n}).
 +
</math>
 +
</center>
 +
 +
Prove that there exists exactly one value of <math>x_1</math> which gives <math>0 < x_n < x_{n + 1} < 1</math> for all <math>n</math>.
  
 
[[1985 IMO Problems/Problem 6 | Solution]]
 
[[1985 IMO Problems/Problem 6 | Solution]]
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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]]}}

Latest revision as of 02:00, 29 March 2021

Problems of the 26th IMO Finland.

Day I

Problem 1

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD + BC = AB$.

Solution

Problem 2

Let $n$ and $k$ be given relatively prime natural numbers, $k < n$. Each number in the set $M = \{ 1,2, \ldots , n-1 \}$ is colored either blue or white. It is given that

(i) for each $i \in M$, both $i$ and $n-i$ have the same color;

(ii) for each $i \in M, i \neq k$, both $i$ and $|i-k|$ have the same color.

Prove that all the numbers in $M$ have the same color.

Solution

Problem 3

For any polynomial $P(x) = a_0 + a_1 x + \cdots + a_k x^k$ with integer coefficients, the number of coefficients which are odd is denoted by $w(P)$. For $i = 0, 1, \ldots$, let $Q_i (x) = (1+x)^i$. Prove that if $i_1, i_2, \ldots , i_n$ are integers such that $0 \leq i_1 < i_2 < \cdots < i_n$, then

$w(Q_{i_1} + Q_{i_2} + \cdots + Q_{i_n}) \ge w(Q_{i_1})$.

Solution

Day II

Problem 4

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer.

Solution

Problem 5

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB = 90^{\circ}$.

Solution

Problem 6

For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting:

$x_{n + 1} = x_n(x_n + {1\over n}).$

Prove that there exists exactly one value of $x_1$ which gives $0 < x_n < x_{n + 1} < 1$ for all $n$.

Solution

Resources

1985 IMO (Problems) • Resources
Preceded by
1984 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1986 IMO
All IMO Problems and Solutions
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