2009 Albania Team Selection Test

Part of Albania Team Selection Test

Subcontests

(4)

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Albania IMO TST 2009 Question 2

Find all the functions

f :\mathbb{R}\mapsto\mathbb{R}

with the following property:

\forall x

f(x)= f(x/2) + (x/2)f'(x)

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Albania IMO TST 2009 Question 4

Find all the natural numbers

m,n

1+5 \cdot 2^m=n^2

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Albania IMO TST 2009 Question 3

Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally which couples of points can they have?

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Albania IMO TST 2009 Question 1

An equilateral triangle has inside it a point with distances 5,12,13 from the vertices . Find its side.