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Difference between revisions of "2007 BMO Problems"

 
(wording; links)
 
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== Problem 1 ==
 
== Problem 1 ==
  
−
Let <math> \displaystyle ABCD </math> be a convex quadrilateral with <math> \displaystyle AB=BC=CD </math> and <math> \displaystyle AC </math> not equal to <math> \displaystyle BD </math>, and let <math> \displaystyle E </math> be the intersection point of its diagonals. Prove that <math> \displaystyle AE=DE </math> if and only if <math> \angle BAD+\angle ADC = 120^{\circ} </math>.
+
Let <math> \displaystyle ABCD </math> be a convex quadrilateral with <math> \displaystyle AB=BC=CD </math>, <math> \displaystyle AC </math> not equal to <math> \displaystyle BD </math>, and let <math> \displaystyle E </math> be the intersection point of its diagonals. Prove that <math> \displaystyle AE=DE </math> if and only if <math> \angle BAD+\angle ADC = 120^{\circ} </math>.
  
 
[[2007 BMO Problems/Problem 1 | Solution]]
 
[[2007 BMO Problems/Problem 1 | Solution]]
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<math>
 
<math>
 
\displaystyle f(f(x) + y) = f(f(x) - y) + 4f(x)y
 
\displaystyle f(f(x) + y) = f(f(x) - y) + 4f(x)y
−
</math>.
+
</math>, for any <math> x,y \in \mathbb{R} </math>.
 
</center>
 
</center>
  
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<center>
 
<center>
 
<math>
 
<math>
−
\sqrt{\sigma(1) + \sqrt{\sigma(2) + \sqrt{ \cdots + \sqrt{\sigma(n-1) + \sqrt{\sigma(n)}}}}}
+
\sqrt{\sigma(1) + \sqrt{\sigma(2) + \sqrt{ \cdots + \sqrt{\sigma(n)}}}}
 
</math>
 
</math>
 
</center>
 
</center>
 
is a rational number.
 
is a rational number.
 +
 +
'''''Note''':  A permutation of the set <math> \{ 1, 2, \ldots, n \} </math> is a one-to-one function of this set to itself.''
  
 
[[2007 BMO Problems/Problem 3 | Solution]]
 
[[2007 BMO Problems/Problem 3 | Solution]]
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== Problem 4 ==
 
== Problem 4 ==
  
−
For a given positive integer <math> \displaystyle n >2 </math>, let <math> \displaystyle C_{1},C_{2},C_{3} </math> be the boundaries of three convex <math> \displaystyle n</math>-gons in the plane such that <math> C_{1}\cap C_{2} </math>, <math> C_{2}\cap C_{3} </math>, <math> C_{1}\cap C_{3} </math> are finite. Find the maximum number of points in the set <math> C_{1}\cap C_{2}\cap C_{3} </math>.
+
For a given positive integer <math> \displaystyle n >2 </math>, let <math> \displaystyle C_{1},C_{2},C_{3} </math> be the boundaries of three convex <math> \displaystyle n</math>-gons in the plane such that <math> C_{1}\cap C_{2} </math>, <math> C_{2}\cap C_{3} </math>, <math> C_{3}\cap C_{1} </math> are finite. Find the maximum number of points in the set <math> C_{1}\cap C_{2}\cap C_{3} </math>.
  
 
[[2007 BMO Problems/Problem 4 | Solution]]
 
[[2007 BMO Problems/Problem 4 | Solution]]
Line 38: Line 40:
 
== Resources ==
 
== Resources ==
  
 +
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=19&year=2007 2007 Balkan MO Problems on the Resources page]
 +
* [http://www.hms.gr/bmo2007/probl.pdf 2007 BMO Problems]
 +
* [http://www.hms.gr/bmo2007/problems.html 2007 BMO Problems and Solutions]
 
* [[Balkan Mathematical Olympiad Problems and Solutions]]
 
* [[Balkan Mathematical Olympiad Problems and Solutions]]

Latest revision as of 23:44, 4 May 2007

Problems of the 2007 Balkan Mathematical Olympiad.

Problem 1

Let $\displaystyle ABCD$ be a convex quadrilateral with $\displaystyle AB=BC=CD$, $\displaystyle AC$ not equal to $\displaystyle BD$, and let $\displaystyle E$ be the intersection point of its diagonals. Prove that $\displaystyle AE=DE$ if and only if $\angle BAD+\angle ADC = 120^{\circ}$.

Solution

Problem 2

Find all functions $\displaystyle f : \mathbb{R} \mapsto \mathbb{R}$ such that

$\displaystyle f(f(x) + y) = f(f(x) - y) + 4f(x)y$, for any $x,y \in \mathbb{R}$.

Solution

Problem 3

Find all positive integers $\displaystyle n$ such that there exists a permutation $\displaystyle \sigma$ on the set $\{ 1, \ldots, n \}$ for which

$\sqrt{\sigma(1) + \sqrt{\sigma(2) + \sqrt{ \cdots + \sqrt{\sigma(n)}}}}$

is a rational number.

Note: A permutation of the set $\{ 1, 2, \ldots, n \}$ is a one-to-one function of this set to itself.

Solution

Problem 4

For a given positive integer $\displaystyle n >2$, let $\displaystyle C_{1},C_{2},C_{3}$ be the boundaries of three convex $\displaystyle n$-gons in the plane such that $C_{1}\cap C_{2}$, $C_{2}\cap C_{3}$, $C_{3}\cap C_{1}$ are finite. Find the maximum number of points in the set $C_{1}\cap C_{2}\cap C_{3}$.

Solution

Resources

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