Difference between revisions of "2007 BMO Problems"
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== Problem 1 == | == Problem 1 == | ||
− | Let <math> \displaystyle ABCD </math> be a convex quadrilateral with <math> \displaystyle AB=BC=CD </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 | + | \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_{ | + | 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]] | ||
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== 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 22:44, 4 May 2007
Problems of the 2007 Balkan Mathematical Olympiad.
Problem 1
Let be a convex quadrilateral with , not equal to , and let be the intersection point of its diagonals. Prove that if and only if .
Problem 2
Find all functions such that
, for any .
Problem 3
Find all positive integers such that there exists a permutation on the set for which
is a rational number.
Note: A permutation of the set is a one-to-one function of this set to itself.
Problem 4
For a given positive integer , let be the boundaries of three convex -gons in the plane such that , , are finite. Find the maximum number of points in the set .