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Difference between revisions of "1987 IMO Problems"

 
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=== Problem 1 ===
 
=== Problem 1 ===
  
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Let <math> \displaystyle p_n (k) </math> be the number of permutations of the set <math> \displaystyle \{ 1, \ldots , n \} , \; n \ge 1 </math>, which have exactly <math> \displaystyle k </math> fixed points.  Prove that
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Let <math>p_n (k) </math> be the number of permutations of the set <math>\{ 1, \ldots , n \} , \; n \ge 1 </math>, which have exactly <math>k </math> fixed points.  Prove that
  
 
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(Remark: A permutation <math> \displaystyle f </math> of a set <math> \displaystyle S </math> is a one-to-one mapping of <math> \displaystyle S </math> onto itself.  An element <math> \displaystyle i </math> in <math> \displaystyle S </math> is called a fixed point of the permutation <math> \displaystyle f </math> if <math> \displaystyle f(i) = i </math>.)
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(Remark: A permutation <math>f </math> of a set <math>S </math> is a one-to-one mapping of <math>S </math> onto itself.  An element <math>i </math> in <math>S </math> is called a fixed point of the permutation <math>f </math> if <math>f(i) = i </math>.)
  
 
[[1987 IMO Problems/Problem 1 | Solution]]
 
[[1987 IMO Problems/Problem 1 | Solution]]
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=== Problem 2 ===
 
=== Problem 2 ===
  
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In an acute-angled triangle <math> \displaystyle ABC </math> the interior bisector of the angle <math> \displaystyle A </math> intersects <math> \displaystyle BC </math> at <math> \displaystyle L </math> and intersects the [[circumcircle]] of <math> \displaystyle ABC </math> again at <math> \displaystyle N </math>. From point <math> \displaystyle L </math> perpendiculars are drawn to <math> \displaystyle AB </math> and <math> \displaystyle AC </math>, the feet of these perpendiculars being <math> \displaystyle K </math> and <math> \displaystyle M </math> respectively.  Prove that the quadrilateral <math> \displaystyle AKNM </math> and the triangle <math> \displaystyle ABC </math> have equal areas.
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In an acute-angled triangle <math>ABC </math> the interior bisector of the angle <math>A </math> intersects <math>BC </math> at <math>L </math> and intersects the [[circumcircle]] of <math>ABC </math> again at <math>N </math>. From point <math>L </math> perpendiculars are drawn to <math>AB </math> and <math>AC </math>, the feet of these perpendiculars being <math>K </math> and <math>M </math> respectively.  Prove that the quadrilateral <math>AKNM </math> and the triangle <math>ABC </math> have equal areas.  
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[[1987 IMO Problems/Problem 2 | Solution]]
 
[[1987 IMO Problems/Problem 2 | Solution]]
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=== Problem 3 ===
 
=== Problem 3 ===
  
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Let <math> \displaystyle x_1 , x_2 , \ldots , x_n </math> be real numbers satisfying <math> \displaystyle x_1^2 + x_2^2 + \cdots + x_n^2 = 1 </math>.  Prove that for every integer <math> \displaystyle k \ge 2 </math> there are integers <math> \displaystyle a_1, a_2, \ldots a_n </math>, not all 0, such that <math> \displaystyle | a_i | \le k-1 </math> for all <math> \displaystyle i </math> and
+
Let <math>x_1 , x_2 , \ldots , x_n </math> be real numbers satisfying <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 1 </math>.  Prove that for every integer <math>k \ge 2 </math> there are integers <math>a_1, a_2, \ldots a_n </math>, not all 0, such that <math>| a_i | \le k-1 </math> for all <math>i </math> and
  
 
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=== Problem 4 ===
 
=== Problem 4 ===
  
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Prove that there is no function <math> \displaystyle f </math> from the set of non-negative  integers into itself such that <math> \displaystyle f(f(n)) = n + 1987 </math> for every <math> \displaystyle n </math>.
+
Prove that there is no function <math>f </math> from the set of non-negative  integers into itself such that <math>f(f(n)) = n + 1987 </math> for every <math>n </math>.
  
 
[[1987 IMO Problems/Problem 4 | Solution]]
 
[[1987 IMO Problems/Problem 4 | Solution]]
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=== Problem 5 ===
 
=== Problem 5 ===
  
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Let <math> \displaystyle n </math> be an integer greater than or equal to 3.  Prove that there is a set of <math> \displaystyle n </math> points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
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Let <math>n </math> be an integer greater than or equal to 3.  Prove that there is a set of <math>n </math> points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
  
 
[[1987 IMO Problems/Problem 5 | Solution]]
 
[[1987 IMO Problems/Problem 5 | Solution]]
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=== Problem 6 ===
 
=== Problem 6 ===
  
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Let <math> \displaystyle n </math> be an integer greater than or equal to 2.  Prove that if <math> \displaystyle k^2 + k + n </math> is prime for all integers <math> \displaystyle k </math> such that <math> 0 \leq k \leq \sqrt{n/3} </math>, then <math> \displaystyle k^2 + k + n </math> is prime for all integers <math> \displaystyle k </math> such that <math> \displaystyle 0 \leq k \leq n - 2 </math>.
+
Let <math>n </math> be an integer greater than or equal to 2.  Prove that if <math>k^2 + k + n </math> is prime for all integers <math>k </math> such that <math> 0 \leq k \leq \sqrt{n/3} </math>, then <math>k^2 + k + n </math> is prime for all integers <math>k </math> such that <math>0 \leq k \leq n - 2 </math>.
  
 
[[1987 IMO Problems/Problem 6 | Solution]]
 
[[1987 IMO Problems/Problem 6 | Solution]]
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* [[1987 IMO]]
 
* [[1987 IMO]]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1987 IMO 1987 problems on the Resources page]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1987 IMO 1987 problems on the Resources page]
 +
* [[IMO Problems and Solutions, with authors]]
 +
* [[Mathematics competition resources]] {{IMO box|year=1987|before=[[1986 IMO]]|after=[[1988 IMO]]}}

Latest revision as of 13:24, 17 September 2023

Problems of the 1987 IMO Cuba.

Day I

Problem 1

Let $p_n (k)$ be the number of permutations of the set $\{ 1, \ldots , n \} , \; n \ge 1$, which have exactly $k$ fixed points. Prove that

$\sum_{k=0}^{n} k \cdot p_n (k) = n!$.

(Remark: A permutation $f$ of a set $S$ is a one-to-one mapping of $S$ onto itself. An element $i$ in $S$ is called a fixed point of the permutation $f$ if $f(i) = i$.)

Solution

Problem 2

In an acute-angled triangle $ABC$ the interior bisector of the angle $A$ intersects $BC$ at $L$ and intersects the circumcircle of $ABC$ again at $N$. From point $L$ perpendiculars are drawn to $AB$ and $AC$, the feet of these perpendiculars being $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.


Solution

Problem 3

Let $x_1 , x_2 , \ldots , x_n$ be real numbers satisfying $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$. Prove that for every integer $k \ge 2$ there are integers $a_1, a_2, \ldots a_n$, not all 0, such that $| a_i | \le k-1$ for all $i$ and

$|a_1x_1 + a_2x_2 + \cdots + a_nx_n| \le \frac{ (k-1) \sqrt{n} }{ k^n - 1 }$.

Solution

Day 2

Problem 4

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$.

Solution

Problem 5

Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

Solution

Problem 6

Let $n$ be an integer greater than or equal to 2. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{n/3}$, then $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

Solution

Resources

1987 IMO (Problems) • Resources
Preceded by
1986 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1988 IMO
All IMO Problems and Solutions
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