Chinese IMO Team Selection Test 2001Time: 4.5 hours each day.

First Day

1. A convex quadrilateral ABCD is given in the plane. Suppose there exist pointsE,F inside the quadrilateral such that

AE = BE, CE = DE, AEB = CED; andAF = DF, BF = CF, AFD = BFC.

Prove that AFD+AEB = pi .

2. Let a,b be integers with b > a > 1 such that a does not divide b, and let (bn)n=1be a sequence of natural numbers such that bn+1 2bn for all n. Does therenecessarily exist a sequence (an)n=1 of natural numbers such that an+1 an {a,b} for all n and am + al 6 (bn)n=1 for any indices m, l?

3. Let k 1 be a given integer. Find all functions f : R R such thatf (xk + f (y)) = y + f (x)k for all x,y R.

Second Day

4. Let n 3 be an integer and let 0 < x1 < x2 < < xn+2 be real numbers. Findthe minimum possible value of

(ni=1 xi+1xi

)(nj=1 x j+2x j+1

)(

nk=1 xk+1xk+2x2k+1+xkxk+2)(

nl=1x2l+1+xlxl+2

xl xl+1

) ,

and determine when this minimum value is attained.

5. Let D be an arbitrary point on the side BC of an equilateral triangle ABC. LetO1, I1 be the circumcenter and incenter of ABD and O2, I2 be the circumcenterand incenter ofACD, respectively. Lines O1I1 and O2I2 meet at a point P. Findthe locus of P as D moves along the segment BC.

6. Find positive real numbers a,b,c for which F = max1x3

|x3ax2bx c| is min-imum possible.

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