Chinese IMO Team Selection Test 2000Time: 4.5 hours each day.

First Day

1. Let ABC be a triangle with AB = AC. Let D,E be points on AB,AC respectivelysuch that DE = AC. Line DE meets the circumcircle of triangle ABC at point T .Let P be a point on AT . Prove that PD + PE = AT if and only if P lies on thecircumcircle of triangle ADE .

2. Given positive integers k,m,n with 1 k m n, evaluaten

i=0

1n + k + i

(m+ n + i)!i!(n i)!(m+ i)!

.

3. For an integer a 2, let Na denote the number of positive integers k with thefollowing property: the sum of squares of digits of k in base a representationequals k. Prove that:

(a) Na is odd;(b) For every M N, there exists an integer a 2 such that Na M.

Second Day

4. Let F be the set of all polynomials (x) with integer coefficients such that theequation (x) = 1 has integer roots. Given a positive integer k, find the smallestinteger m(k) > 1 such that there exists F for which (x) = m(k) has exactlyk distinct integer roots.

5. (a) Let a,b be real numbers. Define the sequences (xn) and (yn) by x0 = 1,y0 = 0 and xk+1 = axk byl, yk+1 = xk ayk for k = 0,1,2, ... Prove that

xk =[k/2]

l=0

(1)lak2l(a2 + b)lk,l,

where k,l =[k/2]

m=l

(k

2m

)(m

l

).

(b) Let uk =[k/2]

l=0

k,l . For a fixed positive integer m, denote the remainder of

uk divided by 2m as zm,k. Prove that the sequence (zm,k), k = 0,1,2, ... isperiodic, and find the smallest period.

6. Given a positive integer n, let us denote M = {(x,y)Z | 1 x,y n}. Considerall functions f : M Z which satisfy:

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

(i) f (x,y) 0 for all x,y;(ii) ny=1 f (x,y) = n1 for each x = 1, . . . ,n;

(iii) if f (x1,y1) f (x2,y2) > 0 then (x1 x2)(y1 y2) 0.Find the number N(n) of functions with the above properties. Determine thevalue of N(4).

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