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Home » Olympiad Resources » International Competitions » Balkan MO » 2007
 
Balkan MO 2007
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27 April 2007

1 Let ABCD a convex quadrilateral with AB=BC=CD, with AC not equal to BD and E be the intersection point of it's diagonals. Prove that AE=DE if and only if \angle BAD+\angle ADC = 120. S
2 Find all real functions f defined on IR, such that f(f(x)+y) = f(f(x)-y)+4f(x)y , for all real numbers x,y. S
3 Find all positive integers n such that there exist a permutation \sigma on the set \{1,2,3, \ldots, n\} for which
\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}
is a rational number.
4 For a given positive integer n >2, let C_{1},C_{2},C_{3} be the boundaries of three convex n- gons in the plane , such that
C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3} are finite. Find the maximum number of points of the sets C_{1}\cap C_{2}\cap C_{3}.
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