Chinese IMO Team Selection Test 2008Time: 4.5 hours each day.

First Day

1. Let ABC be a triangle such that AB > AC. Let E be the point of tangency of BCwith the incircle of ABC. Let D be the second intersection point of the incirclewith the segment AE . Point F AE (F 6= E) satisfies CE = CF . The ray CFintersects BD at G. Prove that CF = FG.

2. The sequence (xn) is defined by x1 = 2, x2 = 12, and xn+2 = 6xn+1 xn, forn 1. Let p be an odd prime number and let q be a prime divisor of xp. Provethat if q 6 {2,3} then q 2p1.

3. Suppose that every positive integer has been painted in one of the two colors.Prove that there exists an infinite sequence of positive integers a1 < a2 < such that a1, a1+a22 ,a2,

a2+a32 ,a3, . . . is an infinite sequence of positive integers of

the same color.

Second Day

4. Let n 4 be an integer. Consider all the subsets of Gn = {1,2, . . . ,n} with atleast two elements. Prove it is possible to arrange those subsets in a sequenceP1,P2, . . . ,P2nn1 such that |PiPi+1|= 2 for every i = 1,2, . . . ,2nn2.

5. Let m,n be two positive integers. Positive real numbers ai, j (1 i n,1 jm)are not all equal to zero. If

f =nni=1

(mj=1 ai j

)2+ mmj=1 (ni=1 ai j)2

(ni=1 mj=1 ai j

)2+ mnni=1 mj=1 a2i j

,

find the maximum and minimum of f .6. Find the maximal constant M such that for arbitrary integer n 3, there exists

two sequences of positive real numbers a1, . . . ,an and b1, . . . ,bn such that

(i) nk=1 bk = 1, 2bk bk1 + bk+1, k = 2,3, . . . ,n1;(ii) a2k 1 + ki=1 aibi, k = 1,2, . . . ,n, an = M.

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The IMO Compendium Group,D. Djukic, V. Jankovic, I. Matic, N. Petrovicwww.imomath.com