8/7/2019 China-Team_Selection_Test-1997-47-37

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ChinaTeam Selection Test

1997

Day 1

1 Given a real number > 1, let P be a point on the arc BAC of the circumcircle ofABC.Extend BP and CP to U and V respectively such that BU = BA, CV = CA. Thenextend U V to Q such that U Q = U V. Find the locus of point Q.

2 There are n football teams in a round-robin competition where every 2 teams meet once. Thewinner of each match receives 3 points while the loser receives 0 points. In the case of a draw,both teams receive 1 point each. Let k be as follows: 2 k n 1. At least how manypoints must a certain team get in the competition so as to ensure that there are at most k1teams whose scores are not less than that particular teams score?

3 Prove that there exists m N such that there exists an integral sequence {an} which satisfies:

I. a0 = 1, a1 = 337;

II. (an+1an1 a2n) +

3

4(an+1 + an1 2an) = m, n 1;

III.1

6(an + 1)(2an + 1) is a perfect square n 1.

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8/7/2019 China-Team_Selection_Test-1997-47-37

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ChinaTeam Selection Test

1997

Day 2

1 Find all real-coefficient polynomials f(x) which satisfy the following conditions:

I. f(x) = a0x2n + a2x

2n2 + + a2n2x2 + a2n, a0 > 0; II.

nj=0

a2ja2n2j

2nn

a0a2n;

III. All the roots of f(x) are imaginary numbers with no real part.

Corrected due to the courtesy of[url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url]

2 Let n be a natural number greater than 6. X is a set such that |X| = n. A1, A2, . . . , Am are

distinct 5-element subsets of X. If m >n(n 1)(n 2)(n 3)(4n 15)

600, prove that there

exists Ai1 , Ai2 , . . . , Ai6 (1 i1 < i2 < , i6 m), such that6

k=1

Aik = 6.

3 There are 1997 pieces of medicine. Three bottles A , B , C can contain at most 1997, 97, 17pieces of medicine respectively. At first, all 1997 pieces are placed in bottle A, and the threebottles are closed. Each piece of medicine can be split into 100 part. When a bottle isopened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all

the medicine. However, he can only open each of the bottles at most once each day, consumeone piece of medicine, move some pieces between the bottles, and close them. At least howmany parts will be lost by the time he finishes consuming all the medicine?

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