8/12/2019 China Team Selection Test 1986 47

1/2

ChinaTeam Selection Test

1986

Day 1

1 If ABCD is a cyclic quadrilateral, then prove that the incenters of the triangles ABC , BCD ,CDA , DAB are the vertices of a rectangle.

2 Let a1 , a2 , ..., an and b1 , b2 , ..., bn be 2 n real numbers. Prove that the following twostatements are equivalent:

i) For any n real numbers x 1 , x 2 , ..., x n satisfying x 1 x 2 . . . x n , we have nk =1 a k x k nk =1 bk x k ,

ii) We have sk =1 a k sk =1 bk for every s {1, 2, . . . ,n 1} and

nk =1 a k = nk =1 bk .

3 Given a positive integer A written in decimal expansion: ( a n , a n 1 , . . . , a 0 ) and let f (A)denote nk =0 2

n k a k . Dene A1 = f (A), A 2 = f (A1 ). Prove that:

I. There exists positive integer k for which Ak +1 = A k . II. Find such Ak for 1986 .

4 Given a triangle ABC for which C = 90 degrees, prove that given n points inside it, we canname them P 1 , P 2 , . . . , P n in some way such that:

n 1k =1 (P K P k +1 )

2 AB 2 (the sum is over the consecutive square of the segments from 1 upto n 1).

Edited by orl.

http://www.artofproblemsolving.com/This le was downloaded from the AoPS Math Olympiad Resources Page Page 1

http://www.artofproblemsolving.com/Forum/resources.php?c=37http://www.mathlinks.ro/Forum/resources.php?c=37&cid=47http://www.artofproblemsolving.com/Forum/resources.php?c=37&cid=47&year=1986http://www.artofproblemsolving.com/Forum/viewtopic.php?p=154713http://www.artofproblemsolving.com/Forum/viewtopic.php?p=234015http://www.artofproblemsolving.com/Forum/viewtopic.php?p=234060http://www.artofproblemsolving.com/Forum/viewtopic.php?p=234065http://www.artofproblemsolving.com/http://www.artofproblemsolving.com/http://www.artofproblemsolving.com/Forum/viewtopic.php?p=234065http://www.artofproblemsolving.com/Forum/viewtopic.php?p=234060http://www.artofproblemsolving.com/Forum/viewtopic.php?p=234015http://www.artofproblemsolving.com/Forum/viewtopic.php?p=154713http://www.artofproblemsolving.com/Forum/resources.php?c=37&cid=47&year=1986http://www.mathlinks.ro/Forum/resources.php?c=37&cid=47http://www.artofproblemsolving.com/Forum/resources.php?c=37

8/12/2019 China Team Selection Test 1986 47

2/2

ChinaTeam Selection Test

1986

Day 2

1 Given a square ABCD whose side length is 1, P and Q are points on the sides AB and AD .If the perimeter of AP Q is 2 nd the angle P CQ .

2 Given a tetrahedron ABCD , E , F , G, are on the respectively on the segments AB , AC andAD . Prove that:

i) area EF G maxarea ABC ,area ABD ,area ACD ,area BCD . ii) The same as abovereplacing area for perimeter.

3 Let x i , 1 i n be real numbers with n 3. Let p and q be their symmetric sum of degree1 and 2 respectively. Prove that:

i) p2 n 1n 2q 0

ii) x i pn p2 2nqn 1 n 1n for every meaningful i.4 Mark 4 k points in a circle and number them arbitrarily with numbers from 1 to 4 k . The

chords cannot share common endpoints, also, the endpoints of these chords should be amongthe 4 k points.

i. Prove that 2 k pairwisely non-intersecting chords can be drawn for each of whom its

endpoints differ in at most 3 k 1. ii. Prove that the 3 k 1 cannot be improved.

http://www.artofproblemsolving.com/This le was downloaded from the AoPS Math Olympiad Resources Page Page 2

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