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:
1
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).
2
The IMO Compendium Group,D. Djukic, V. Jankovic, I. Matic, N.
Petrovicwww.imomath.com
