# 2010_eng - types of operation allowed Type 1 Choose a...

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Wednesday, July 7, 2010 Problem 1. Determine all functions f : R R such that the equality f Ä b x c y ä = f ( x ) ö f ( y ) ù holds for all x,y R . (Here b z c denotes the greatest integer less than or equal to z .) Problem 2. Let I be the incentre of triangle ABC and let Γ be its circumcircle. Let the line AI intersect Γ again at D . Let E be a point on the arc ˙ BDC and F a point on the side BC such that BAF = CAE < 1 2 BAC. Finally, let G be the midpoint of the segment IF . Prove that the lines DG and EI intersect on Γ . Problem 3. Let N be the set of positive integers. Determine all functions g : N N such that Ä g ( m ) + n äÄ m + g ( n ) ä is a perfect square for all m,n N . Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points Language: English Day: 1

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Thursday, July 8, 2010 Problem 4. Let P be a point inside the triangle ABC . The lines AP , BP and CP intersect the circumcircle Γ of triangle ABC again at the points K , L and M respectively. The tangent to Γ at C intersects the line AB at S . Suppose that SC = SP . Prove that MK = ML . Problem 5. In each of six boxes B 1 ,B 2 ,B 3 ,B 4 ,B 5 ,B 6 there is initially one coin. There are two
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Unformatted text preview: types of operation allowed: Type 1: Choose a nonempty box B j with 1 ≤ j ≤ 5 . Remove one coin from B j and add two coins to B j +1 . Type 2: Chooseanonemptybox B k with 1 ≤ k ≤ 4 . Removeonecoinfrom B k andexchange the contents of (possibly empty) boxes B k +1 and B k +2 . Determinewhetherthereisaﬁnitesequenceofsuchoperationsthatresultsinboxes B 1 ,B 2 ,B 3 ,B 4 ,B 5 being empty and box B 6 containing exactly 2010 2010 2010 coins. (Note that a b c = a ( b c ) .) Problem 6. Let a 1 ,a 2 ,a 3 ,... be a sequence of positive real numbers. Suppose that for some positive integer s , we have a n = max { a k + a n-k | 1 ≤ k ≤ n-1 } forall n > s . Provethatthereexistpositiveintegers ` and N ,with ` ≤ s andsuchthat a n = a ` + a n-` for all n ≥ N . Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points Language: English Day: 2...
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2010_eng - types of operation allowed Type 1 Choose a...

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