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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1121.0. "The 30th International Mathematical Olympiad" by HPSTEK::XIA (In my beginning is my end.) Fri Sep 15 1989 19:11

From: Notices of the American Mathematical Society, Sept 1989
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	A team of six American high school students placed fifth in 
	the 30th International Mathematical Olympiad, held July 14-24
	in Braunschweig, West Germany, with one team member receiving a 
	perfect score.

	The American team scored 207 out of a possible 252 and led the 
	participating western countries in a competition dominated by eastern
	nations.  Ahead of the Americans were teams from China (237), Romania
	(223), the U.S.S.R. (217), and East Germany (216).  There were only 3
	western countries among the top 13 teams, with West Germany placing
	8th and France 13th.  Fifty countries sent a total of 291 students to
	the competition.

	The judges also awarded individual first, second, and third prizes
	to deserving team members.  Jordan Ellenberg of Potomac, MD received 
	gold medal for a perfect score of 42, one of only 10 perfect scores.
	Four American team members received silver medals: Samuel Kutin of 
	Old Westbury, NY (32), Andrew Kresch of Havertown, PA (37), Jeffrey
	Vanderkam of Raleigh, NC (35), and Samuel Vandervelde of Amherst, VA 
	(32).  David Carlton of Oberlin, OH received a bronze medal (24).
	Twenty gold, 55 silver, and 72 bronze medals were awarded.
    ----------------------------------------------------------------------------



Eugene

T.R Title User Personal
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1121.1 AUSSIE::GARSON achtentachtig kacheltjes Sun Apr 24 1994 22:50 69

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1121.1		AUSSIE::GARSON	achtentachtig kacheltjes	`Sun Apr 24 1994 22:50`	69
	[DG: Better late than never here are the problems for the 30th IMO (1989).] ================================================================================ English version FIRST DAY Braunschweig, July 18th 1989 1. Prove that the set {1,2,...,1989} can be expressed as the disjoint union of subsets A[i] (i=1,2,...,117) such that (i) each A[i] contains 17 elements, and (ii) the sum of all the elements in each A[i] is the same 2. In an acute-angled triangle ABC the internal bisector of angle A meets the circumcircle of the triangle again at A[1]. The points B[1] and C[1] are defined similarly. Let A[0] be the point of intersection of the line AA[1] with the external bisectors of angles B and C. Points B[0] and C[0] are defined similarly. [DG: What is the external bisector anyway?] Prove that (i) the area of the triangle A[0]B[0]C[0] is twice the area of the hexagon AC[1]BA[1]CB[1] (ii) the area of the triangle A[0]B[0]C[0] is at least four times the area of the triangle ABC. 3. Let n and k be positive integers and let S be a set of n points in the plane such that (i) no three points of S are collinear, and (ii) for every point P of S there are at least k points of S equidistant from P. Prove that k < � + sqrt(2n) Time: 4.5 hours Each problem is worth 7 points. SECOND DAY Braunschweig, July 19th 1989 4. Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC. Show that 1 1 1 ------- >= -------- + -------- sqrt(h) sqrt(AD) sqrt(BC) 5. Prove that for each positive integer n there exist n consecutive positive integers none of which is an integral power of a prime number. 6. A permutation (x[1],x[2],...,x[2n]) of the set {1,2,...,2n}, where n is a positive integer, is said to have property P if \|x[i]-x[i+1]\| = n for at least one i in {1,2,...,2n-1}. Show that for each n, there are more permutations with property P than without. Time: 4.5 hours Each problem is worth 7 points.

[DG: Better late than never here are the problems for the 30th IMO (1989).]
================================================================================
English version

                                  FIRST DAY

                         Braunschweig, July 18th 1989

1. Prove that the set {1,2,...,1989} can be expressed as the disjoint union of
   subsets A[i] (i=1,2,...,117) such that

   (i)  each A[i] contains 17 elements, and
   (ii) the sum of all the elements in each A[i] is the same

2. In an acute-angled triangle ABC the internal bisector of angle A meets the
   circumcircle of the triangle again at A[1]. The points B[1] and C[1] are
   defined similarly. Let A[0] be the point of intersection of the line AA[1]
   with the external bisectors of angles B and C. Points B[0] and C[0] are
   defined similarly.

[DG: What is the external bisector anyway?]

   Prove that

   (i)  the area of the triangle A[0]B[0]C[0] is twice the area of the hexagon
        AC[1]BA[1]CB[1]
   (ii) the area of the triangle A[0]B[0]C[0] is at least four times the area
        of the triangle ABC.

3. Let n and k be positive integers and let S be a set of n points in the plane
   such that

   (i)  no three points of S are collinear, and
   (ii) for every point P of S there are at least k points of S equidistant
        from P.

   Prove that

   k < � + sqrt(2n)

Time: 4.5 hours
Each problem is worth 7 points.

                                  SECOND DAY

                         Braunschweig, July 19th 1989

4. Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy
   AB = AD + BC. There exists a point P inside the quadrilateral at a distance
   h from the line CD such that AP = h + AD and BP = h + BC.

   Show that

	 1           1          1
      -------  >= -------- + --------
      sqrt(h)     sqrt(AD)   sqrt(BC)

5. Prove that for each positive integer n there exist n consecutive positive
   integers none of which is an integral power of a prime number.

6. A permutation (x[1],x[2],...,x[2n]) of the set {1,2,...,2n}, where n is a
   positive integer, is said to have property P if |x[i]-x[i+1]| = n for at
   least one i in {1,2,...,2n-1}.

   Show that for each n, there are more permutations with property P than
   without.

Time: 4.5 hours
Each problem is worth 7 points.