This email is from AMC's archives and the contents contain the information we currently have on the Record breaking six perfect scores for the 1994 US IMO Team.

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From llng@husc Tue Jul 19 21:10:23 1994

Received: by scws7.harvard.edu; Tue, 19 Jul 94 21:12:21 -0400

Date: Tue, 19 Jul 1994 21:11:48 -0400 (EDT)

From: "O fortuna, velut luna statu variabilis"

Subject: IMO '94 stuff (fwd)

To: Jack Ng, Kiran Kedlaya

Subject: IMO '94 stuff (fwd) I'm forwarding this with the same intro as was adressed to me (except I didn't know and you (probably) do). I'm not going to get to the problems within the next couple of days, so tell me if they're easy... I mean, Holy Shit! They must be. A 29 won't get you a silver!?... But still China got wasted...Congratulate everybody you'll talk to for me (well, I mean everybody you'll talk to intersected with tem members). Tim C. ---

Subject: IMO '94 stuff Hey Tim, You've probly already heard all about this, but I saw it on the Usenet feeds & thought I'd forward it to you. It's an article, the problems, and the table of results. Pretty cool I think. Blake --

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From: claird@Starbase.NeoSoft.COM (Cameron Laird) Date: 18 Jul 1994 12:32:09 -0500

I quote from a report fed to me by a representative of the (USA-based) Joint Policy Board for Mathematics: >

US TEAM MAKES HISTORY AT INTERNATIONAL MATHEMATICAL OLYMPIAD Competing against teams representing 69 countries, a team of six American high school students placed first in the 35th International Mathematical Olympiad with six perfect scores. This is the first time in the 35-year history of the Olympiad that any team has achieved a perfect score. Each US team member scored the maximum number of points (42) on the nine-hour competition, which took place July 8-20 in Hong Kong, and each received a gold medal. The members of the team are: Jeremy Bem, Ithaca, NY; Aleksandr Khazanov, New York, NY; Jacob Lurie, Silver Spring, MD; Noam Shazeer, Swampscott, MA; Stephen Wang, Aurora, IL; and Jonathan Weinstein, Lexington, MA. The teams placing second through fifth are from China, Russia, Bulgaria, and Hungary, in that order. The US team was chosen on the basis of performance in the 23rd annual USA Mathematical Olympiad held earlier this year, and then participated in a month-long summer program at the US Naval Academy. The Mathematical Olympiad is a program of the Mathematical Association of America and is co-sponsored by 9 mathematical organizations. Financial support is provided by the Army Research Office, the Office of Naval Research, Microsoft Corporation, and the Matilda R. Wilson Fund. From: eetszmei@uxmail.ust.hk (Tsz-Mei Ko)

Subject: 35th IMO Problems and Results Date: Mon, 18 Jul 1994 12:37:41

GMT Problems for the 35th International Mathematical Olympiad: (held in Hong Kong, July 12-19, 1994) ---------------------------------------------------------

First Day (9am-1:30pm, July 13, 1994; Each problem is worth 7 points)

Problem 1 (proposed by France) Let m and n be positive integers. Let a_1, a_2, ..., a_m be distinct elements of {1, 2, ..., n} such that whenever a_i + a_j le n for some i, j, 1 le i le j le m, there exists k, 1 le k le m, with a_i + a_j = a_k. Prove that a_1 + a_2 + ... + a_m n + 1 ------------------------- ge --------- m 2

Problem 2 (proposed by Armenia/Australia) ABC is an isosceles triangle with AB=AC. Suppose that (i) M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB; (ii) Q is an arbitrary point on the segment BC different from B and C; (iii) E lies on the line AB and F lies on the line AC such that E, Q and F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE=QF.

Problem 3 (proposed by Romania) For any positive integer k, let f(k) be the number of elements in the set {k+1, k+2, ..., 2k} whose base 2 representation has precisely three 1s. (a) Prove that, for each positive integer m, there exists at least one positive integer k such that f(k)=m. (b) Determine all positive integer m for which there exists exactly one k with f(k)=m. --------------------------------------------------------------------------

Second Day (9am-1:30pm, July 14, 1994; Each problem is worth 7 points)

Problem 4 (proposed by Australia) Determine all ordered pairs (m,n) of positive integers such that n^3 + 1 ------------- mn - 1 is an integer.

Problem 5 (proposed by United Kingdom) Let S be the set of real numbers strictly greater than -1. Find all functions f: S -> S satisfying the two conditions: (i) f(x+f(y)+xf(y)) = y+f(x)+yf(x) for all x and y in S; (ii) f(x)/x is strictly increasing on each of the intervals -1 < x < 0 and 0 < x.

Problem 6 (proposed by Finland) Show that there exists a set A of positive integers with the following property: For any infinite set S of primes there exist two positive integers m in A and n otin A each of which is a product of k distinct elements of S for some k ge 2. **************************************************************************** Result of the 35th International Mathematical Olympiad (held in Hong Kong, July 12-19, 1994) Rank Teams Score Gold Silver Bronze

1 United States 252 6 0 0

2 China 229 3 3 0

3 Russia 224 3 2 1

4 Bulgaria 223 3 2 1

5 Hungary 221 1 5 0

6 Vietnam 207 1 5 0

7 United Kingdom 206 2 2 2

8 Iran 203 2 2 2

9 Romania 198 0 5 1

10 Japan 180 1 2 3

11 Australia 173 0 2 3

12 Chinese Taipei 170 0 4 1

12 Republic of Korea 170 0 2 4

14 India 168 0 3 3

15 Ukraine 163 1 1 2

16 Hong Kong 162 0 2 4

17 France 161 1 1 3

18 Poland 160 2 0 3

19 Argentina 159 0 3 1

20 Czech Republic 154 0 2 2

21 Slovakia 150 1 1 2

22 Germany 145 1 2 3

23 Belarus 144 0 1 4

24 Canada 143 1 0 3

24 Israel 143 0 1 4

26 Colombia 136 0 2 2

27 South Africa 120 0 0 3

28 Turkey 118 0 0 4

29 New Zealand 116 0 0 4

29 Singapore 116 0 2 0

31 Austria 114 1 0 0

32 Armenia/5 110 0 0 4

33 Thailand 106 0 0 3

34 Belgium 105 0 0 2

34 Morocco 105 0 0 2

36 Italy 102 0 0 2

37 Netherlands 99 0 0 2

38 Latvia 98 0 0 3

39 Brazil/5 95 0 2 0 3

9 Republic of Georgia 95 0 0 2

41 Sweden 92 0 0 1

42 Greece 91 0 0 1

43 Croatia 90 0 0 2

44 Estonia/5 82 0 0 1

45 Norway 80 0 1 1

46 Macau 75 0 1 0

47 Lithuania 73 0 0 1

48 Finland 70 0 0 0

49 Ireland 68 0 0 0

50 Macedonia/4 67 0 0 1

51 Mongolia 65 0 1 0

52 Trinidad & Tobago 63 0 0 0

53 Philippines 53 0 0 0

54 Chile/2 52 0 1 0

55 Moldova 52 0 0 1

55 Portugal 52 0 0 0

57 Denmark/4 51 0 0 2

58 Cyprus 48 0 0 0

59 Slovenia/5 47 0 0 0

60 Indonesia 46 0 0 0

61 Bosnia/5 44 0 0 1

62 Spain 41 0 0 0

63 Switzerland/3 35 0 0 1

64 Luxembourg/1 32 0 1 0

65 Iceland/4 29 0 0 0

66 Mexico 29 0 0 0

67 Kryghyzstan 24 0 0 0

68 Cuba/1 12 0 0 0

69 Kuwait/5 12 0 0 0

Remarks:

1. A full team consists of 6 students. Armenia/5 means that Armenia sent only 5 students.

2. The maximum possible score is 252 points: 6 students x 6 problems/student x 7 points/problem = 252 points.

3. For Problem #2, the "if" part was proposed by Australia and the "only if" part was proposed by Armenia. It is interesting that two countries that are in two different hemispheres, with different languages, without any communication, would propose the same problem such that their proposed problems complement each other.

Some statistics (from the IMO committee): Number of participating teams: 69 Number of participating contestants: 385 Number of Gold Medals (scored 40-42): 30 Number of Silver Medals(scored 30-39): 64 Number of Bronze Medals(scored 19-29): 98 Future host countries for the IMO: 1995: Canada 1996: India 1997: Argentina 1998: Taiwan 1999: Romania 2000: South Korea

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