Team Selection Tests for Chinese IMO Team 2004

Team Selection Tests for Chinese IMO Team 2004
First Test
8:00 AM - 12:00 AM
March 18, 2004
1. Using AB, AC as diameters, two semi-circles were constructed respectively outside the acute
triangle ABC. AH⊥BC and intersects BC at H, D is any point on side BC (D is not coincide
with B or C), Through D, draw DE∥AC and DF∥AB, intersecting the two semi-circles at E,
F. Prove that D, E, F, and H are concyclic.
2. 21 girls and 20 boys took part in a mathematical competition. It turned out that:
(i) Every contestant solved at most 6 problems;
(ii) For each pair of a girl and a boy, there was at least one problem that was solved by both of
them.
Prove that there exists a problem that was solved at least 3 girls and at least 3 boys.
(Note: not same as IMO 2001-3, there are 20 boys, not 21 boys.)
3. Find all the positive integer n satisfying the following condition:
There exist positive integers m, a 1 , a 2 , …, a m-1 , such that n = a 1 (m - a 1 ) + a 2 (m - a 2 ) + … +
a m-1 (m - a m-1 ), where a 1 , a 2 , …, a m-1 may not distinct and 1 a i m - 1 (i =1, 2, …, m - 1).
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Team Selection Tests for Chinese IMO Team 2004
Second Test
8:00 AM - 12:00 AM
March 19, 2004
4. Given integer n larger than 5, solve the system of functions
x 1 + x 2 + x 3 + … + x n = n + 2,
x 1 + 2x 2 + 3x 3 + … + nx n = 2n + 2,
x 1 + 2 2 x 2 + 3 2 x 3 + … + n 2 x n = n 2 + n + 4,
x 1 + 2 3 x 2 + 3 3 x 3 + … + n 3 x n = n 3 + n + 8,
where x i 0, i = 1, 2, …, n.
5. Convex quadrilateral ABCD is inscribed in a circle, A = 60 o , BC = CD = 1, radials AB, DC
intersect each other at E, radials BC, AD intersect each other at F. Knowing that the perimeters
of triangles BCE and CDF are both integers. Find the perimeter of quadrilateral ABCD.
6. S is a nonempty subset of set {1, 2, …, 108}, satisfying:
(i) For any two numbers a, b S (may not distinct), there exists c S such that gcd(a, c) =
gcd(b, c) = 1;
(ii) For any two numbers a, b S (may not distinct), there exists c' S, c' a, c' b, such that
gcd(a, c') > 1, gcd(b, c') > 1.
Find the largest possible value of |S|.
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Team Selection Tests for Chinese IMO Team 2004
Third Test
14:00 PM - 18:00 PM
March 22, 2004
7. Let m 1 , m 2 , …, m r (may not distinct) and n 1 , n 2 , …, n s (may not distinct) be two groups of
positive integers such that:
For any positive integer d larger than 1, the numbers of which can be divided by d in group m 1 ,
m 2 , …, m r (including repeated numbers, i.e. there are 3 numbers can be divided by 3 among 6,
6, 3, 2) are no less than that of in group n 1 , n 2 , …, n s (including repeated numbers).
Prove that
m1m2
mr
n n n
1
2
s
is integer.
8. Two equal radii circles with centers O 1 and O 2 intersect each other at P, Q, O is the midpoint of
the common chord PQ, Two lines AB and CD are draw through P (AB and CD are not coincide
with PQ) such that A, C lie on the circle O 1 and B, D lie on the circle O 2 , M and N are the
midpoints of segments AD and BC respectively. Knowing that O 1 and O 2 are not in the
common part of the two circles, and M, N are not coincide with O. Prove that M, N, O are
collinear.
9. Given arbitrary positive integer a larger than 1, show that for any positive integer n, it always
exists an n -degree integral coefficient polynomial p(x), such that p(0), p(1), …, p(n) are
pairwise different positive integers, and all have the form of 2a k + 3, where k is integer.
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Team Selection Tests for Chinese IMO Team 2004
Fourth Test
14:00 PM - 18:00 PM
March 23, 2004
10. Find the largest value of real , such that:
As long as point P lies in t he acute triangle ABC satisfying PAB = PBC = PCA, and
radials AP, BP, CP intersect the circumcircle of triangles PBC, PCA, PAB at points A 1 , B 1 , C 1 ,
respectively, then S A1 BC + S B1 CA + S C1 AB S ABC , where S means the area.
11. Find the largest positive real k, such that for any positive reals a, b, c, d, we have
(a + b + c)[3 4 (a + b + c + d) 5 + 2 4 (a + b + c + 2d) 5 ] kabcd 3 .
12. Find all the positive integer m satisfying the following condition: there exits prime p such that
n m - m can not divided by p for any integer n.
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Team Selection Tests for Chinese IMO Team 2004
Fifth Test
8:00 AM - 12:00 AM
March 26, 2004
13. Given nonzero reals a, b, find all the functions f: R R, such that for every x, y R and y 0,
f(2x) = af(x) + bx, and f(x)f(y) = f(xy) +
x
f , where R means the set of real number.
y
14. Let k be a positive integer. Set A Z is called a "k-set" if there exist x 1 , x 2 , …, x k Z such
that for any i j, (x i + A)∩(x j + A) = , where x + A = {x + a | a A}. Prove that if A i are
k i -set (i = 1, 2, …, t), and A 1 ∪A 2 ∪…∪A t = Z, then
1
k
1
1 1
1.
k
2
k t
15. In convex quadrilateral ABCD, AB = a, BC = b, CD = c, DA = d, AC = e, BD = f. If max{a, b,
c, d, e, f} = 1, find the maximum value of abcd.
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Team Selection Tests for Chinese IMO Team 2004
Sixth Test
8:00 AM - 12:00 AM
March 27, 2004
16. Given sequence {c n } satisfying the conditions that c 0 = 1, c 1 = 0, c 2 = 2005, c n+2 = -3c n - 4c n-1 +
2008 (n = 1, 2, 3, …). Let a n = 5(c n+2 - c n )(502 - c n-1 - c n-2 ) + 4 n 2004 501 (n = 2, 3, …). Is
a n is a perfect square for every n > 2 ?
17. There are n 5 pairwisely different points in the plane. For every point, there are just four
points whose distance from it is 1. Find the minimum value of n.
1
2
t
18. The largest one of numbers p , p , …, and
1
2
p
t
is called "good number" of positive
integer n, if n =
1
2
t
p1 p2
p t
, where p 1 , p 2 , …, p t are pairwisely different primes and 1 ,
2 , …, t are positive integers. Let n 1 , n 2 , …, n 10000 be 10000 distinct positive integers such
that the "good numbers" of n 1 , n 2 , …, n 10000 are all equal. Prove that there exist integers a 1 ,
a 2 , …, a 10000 such that any two of following 10000 arithmetical progressions {a i , a i + n i , a i +
2n i , a i + 3n i , …} (i = 1, 2, …, 10000) have no common terms.
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