Imo 2009 (1 Viewer)

buchanan · Jun 28, 2009

The International Mathematical Olympiad 2009 was held on July 15-16 at Jacobs University, Bremen, Germany.

Website: http://www.imo2009.de

Australian Team:

Aaron Chong, Doncaster Secondary College
Andrew Elvey Price, Brunswick Secondary College
Stacey Law, James Ruse Agricultural High School
Alfred Liang, Trinity Grammar School
Dana Ma, Melbourne Girls Grammar School
Sampson Wong, James Ruse Agricultural High School

(From left: Andrew Elvey Price, Aaron Chong, Stacey Law, Margaret Sheil, Alfred Liang, Dana Ma, Sampson Wong)

<a href="http://community.boredofstudies.org/attachments/238/appreciating-beauty-elegance-extracurricular-topics/attachments/imo-2009/18859d1247746453/imo2009.pdf">Here are the Questions</a>

<a href="http://community.boredofstudies.org/4469296/post-13.html">Here are the results</a> (thanks kurt.physics)

Congratulations to Andrew Elvey Price and Sampson Wong for their Golds, Aaron Chong for his Silver, Stacey Law and Alfred Liang for their Bronzes and Dana Ma for her Honourable Mention!

Solutions are supposed to go up on the official website, but aren't there yet. In the meantime, there are some solutions at

http://www.mathlinks.ro/index.php?f=580

<a href="http://terrytao.files.wordpress.com/2009/07/primes.pdf">And here is Terry Tao's Lecture he gave at the IMO.</a>

The next IMO will be in Astana, Kazakhstan, July 6-12, 2010:

http://www.imo2010org.kz/?lang=eng&id_rubric=1

tommykins · Jul 1, 2009

looks like the socceroos team, cept asian.

MaNiElla · Jul 1, 2009

buchanan said:
The International Mathematical Olympiad 2009 will be held on July 15-16 at Jacobs University, Bremen, Germany.

Website: welcome to IMO 2009 Bremen

Australian Team:

Aaron Chong, Doncaster Secondary College
Andrew Elvey Price, Brunswick Secondary College
Stacey Law, James Ruse Agricultural High School
Alfred Liang, Trinity Grammar School
Dana Ma, Melbourne Girls Grammar School
Sampson Wong, James Ruse Agricultural High School

(From left: Andrew Elvey Price, Aaron Chong, Stacey Law, Margaret Sheil, Alfred Liang, Dana Ma, Sampson Wong)

OMG!!! those girls look so "unusuall" imo

babikakez · Jul 1, 2009

Hardly looks like an 'Australian' team.

addikaye03 · Jul 1, 2009

babikakez said:
Hardly looks like an 'Australian' team.

haha, you said what we were all thinking. It's Spot the Aussie.

buchanan · Jul 16, 2009

Here are Questions 1-3.

Question 1.

$Let $n$ be a positive integer and let $a_1,a_2,a_3,\ldots,a_k$ $\left( k\ge 2\right)$ be distinct integers in the set $\left\{ 1,2,\ldots,n\right\}$ such that $n$ divides $a_i(a_{i + 1} - 1)$ for $i = 1,2,\ldots,k - 1$. Prove that $n$ does not divide $a_k(a_1 - 1).$$

Question 2.

$Let $ABC$ be a triangle with circumcentre $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$ respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that $OP = OQ.$$

Question 3.

$Suppose that $s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences $s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1 + 1},s_{s_2 + 1},s_{s_3 + 1},\ldots$ are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3, \ldots$ is itself an arithmetic progression.$

tommykins · Jul 16, 2009

mate is 5:10 am !!! haha.

tommykins · Jul 16, 2009

the tex should be a / , not a \

you also need to fix up the spaces using \text{}

buchanan · Jul 16, 2009

Ta. Yeah. / not \

I also made a pdf.

I'll put Q4-6 up later today.

kurt.physics · Jul 16, 2009

buchanan said:
Here are Questions 1-3.

Question 1.

$Let $n$ be a positive integer and let $a_1,a_2,a_3,\ldots,a_k$ $\left( k\ge 2\right)$ be distinct integers in the set $\left\{ 1,2,\ldots,n\right\}$ such that $n$ divides $a_i(a_{i + 1} - 1)$ for $i = 1,2,\ldots,k - 1$. Prove that $n$ does not divide $a_k(a_1 - 1).$$

Question 2.

$Let $ABC$ be a triangle with circumcentre $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$ respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that $OP = OQ.$$

Question 3.

$Suppose that $s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences $s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1 + 1},s_{s_2 + 1},s_{s_3 + 1},\ldots$ are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3, \ldots$ is itself an arithmetic progression.$

PDF attached below. I will update it later with Q4-6.

Just to add to this:

Problem 1 proposed by Ross Atkins, Australia

Problem 2 proposed by Sergei Berlov, Russia

Problem 3 proposed by Gabriel Carroll, USA

buchanan · Jul 16, 2009

Questions 4-6:

Question 4.

$Let $ABC$ be a triangle with $AB = AC$ . The angle bisectors of $\angle C AB$ and $\angle AB C$ meet the sides $B C$ and $C A$ at $D$ and $E$ , respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle B E K = 45^\circ$ . Find all possible values of $\angle C AB$ .$

Question 5.

$Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $$a, f(b) \text{ and } f(b + f(a) - 1).$$(A triangle is non-degenerate if its vertices are not collinear.)$

Question 6.

$Let $a_1, a_2, \ldots , a_n$ be distinct positive integers and let $M$ be a set of $n - 1$ positive integers not containing $s = a_1 + a_2 + \ldots + a_n.$ A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M.$$

Updated PDF file with Q1-6:

kurt.physics · Jul 20, 2009

buchanan said:
Questions 4-6:

Question 4.

$Let $ABC$ be a triangle with $AB = AC$ . The angle bisectors of $\angle C AB$ and $\angle AB C$ meet the sides $B C$ and $C A$ at $D$ and $E$ , respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle B E K = 45^\circ$ . Find all possible values of $\angle C AB$ .$

Question 5.

$Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $$a, f(b) \text{ and } f(b + f(a) - 1).$$(A triangle is non-degenerate if its vertices are not collinear.)$

Question 6.

$Let $a_1, a_2, \ldots , a_n$ be distinct positive integers and let $M$ be a set of $n - 1$ positive integers not containing $s = a_1 + a_2 + \ldots + a_n.$ A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M.$$

Updated PDF file with Q1-6:

Problem 4 proposed by Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea

Problem 5 proposed by Bruno Le Floch, France

Problem 6 proposed by Dmitry Khramtsov, Russia

kurt.physics · Jul 20, 2009

Results =)

Official 2009 IMO site welcome to IMO 2009 Bremen

Official IMO site International Mathematical Olympiad

Australian Mathematics Trust site Australian Mathematics Trust

Cut-Offs for Bronze, Silver and Gold were 14, 22 and 32 points respectively

China came first with 6 gold medals (the only team to get 6 gold), Japan came second and Russia came third.

There were 2 perfect scores made by Dongyi Wei from the Chinese team:

This is in fact his second perfect score at the IMO (he has been only twice!!!)

And the second perfect score was made by Makoto Soejima of the Japanese team:

This is his fourth IMO!!! He got a Bronze in 2005, a Gold in 2007 and 2008 AND a perfect score and Gold this year.

Australia came 23rd with 2 Golds!

Andrew Elvey Price practically scored a perfect score (but just missing a few silly points), putting him 10th in the IMO this year and winning a Gold medal. This is out of the 545 best high school mathematics students in the world!

Sampson Wong followed with winning a Gold medal ranking him 27th in the Olympiad. Aaron Chong won himself a Silver medal. Both Alfred Liang and Stacey Law achieve Bronze and Dana Ma got an honorable mention.

There results are as follows:

Contestant P1 P2 P3 P4 P5 P6 Total Rank Award

Andrew Elvey Price 7 6 7 5 7 5 37 10 Gold
Sampson Wong 7 7 7 7 6 0 34 27 Gold
Aaron Chong 7 7 7 2 7 0 30 54 Silver
Stacey Law 5 7 0 7 3 0 22 168 Bronze
Alfred Liang 7 3 0 0 7 0 17 220 Bronze
Dana Ma 7 4 0 0 0 0 11 314 Honourable mention

Here is a picture of the Team with Terry Tao.

(from left to right: Dana Ma, Stacey Law, Andrew Elvey Price, Aaron Chong, Terrence Tao, Alfred Liang and Sampson Wong)

This is an amazing result for Australia and we all should be proud!

Here are the team ranks for the 2009 IMO followed by their total score:

1. People's Republic of China 221
2. Japan 212
3. Russian Federation 203
4. Republic of Korea 188
5. Democratic People's Republic of Korea 183
6. United States of America 182
7. Thailand 181
8. Turkey 177
9. Germany 171
10. Belarus 167
11. Taiwan 165
11. Italy 165
13. Romania 163
14. Ukraine 162
15. Vietnam 161
15. Islamic Republic of Iran 161
17. Brazil 160
18. Canada 158
19. United Kingdom 157
19. Bulgaria 157
19. Hungary 157
22. Serbia 153
23. Australia 151
24. Peru 144
25. Georgia 140
25. Poland 140
27. Kazakhstan 136
28. India 130
29. Hong Kong 122
30. Singapore 116
31. France 112
32. Croatia 110
33. Portugal 99
34. Turkmenistan 97
35. Argentina 93
36. The former Yugoslav Republic of Macedonia 91
36. Azerbaijan 91
38. Belgium 89
39. Colombia 88
40. Czech Republic 87
41. Greece 86
42. Uzbekistan 85
43. Indonesia 84
43. South Africa 84
45. Tajikistan 82
46. Israel 80
47. Netherlands 79
47. Switzerland 79
49. Lithuania 77
50. Mexico 74
50. Republic of Moldova 74
50. Sri Lanka 74
53. Slovakia 73
54. Mongolia 72
55. Spain 71
56. Sweden 70
57. Denmark 68
58. Bangladesh 67
59. Austria 66
60. Luxembourg 65
61. Bosnia and Herzegovina 63
62. Latvia 61
63. Norway 60
64. Armenia 59
65. Slovenia 58
66. New Zealand 53
67. Macau 49
67. Finland 49
69. Cyprus 45
70. Chile 41
71. Estonia 40
72. Costa Rica 34
73. Kyrgyzstan 33
74. Morocco 32
75. Malaysia 31
76. Trinidad and Tobago 28
77. Tunisia 27
78. Ecuador 26
78. Iceland 26
78. Philippines 26
81. Albania 24
81. Honduras 24
83. Montenegro 23
83. Puerto Rico 23
85. Pakistan 21
85. Liechtenstein 21
85. Uruguay 21
85. Cuba 21
89. Ireland 20
90. Nigeria 17
91. Cambodia 14
91. Paraguay 14
91 Guatemala 14
94. Venezuela 13
94. El Salvador 13
96. Panama 12
97. Bolivia 9
98. Mauritania 8
99. Syria 7
100. Zimbabwe 5
101. Benin 3
101. United Arab Emirates 3
101. Kuwait 3
104. Algeria 104

addikaye03 · Jul 20, 2009

Re: Results =)

kurt.physics said:
Andrew Elvey Price practically scored a perfect score (but just missing a few silly points), putting him 10th in the IMO this year and winning a Gold medal. This is out of the 545 best high school mathematics students in the world!

WOW! This kid is a freak,

duckcowhybrid · Jul 20, 2009

The only white kid is the smartest. Thats how it always works. He's probably a pure genius.

khorne · Jul 20, 2009

duckcowhybrid said:
The only white kid is the smartest. Thats how it always works. He's probably a pure genius.

or russian...It's practically in their genes.

kurt.physics · Jul 20, 2009

duckcowhybrid said:
The only white kid is the smartest. Thats how it always works. He's probably a pure genius.

Add to this a Silver medal at the 2008 IMO, a Gold and 2 Silver Asian Pacific Mathematical Olympiad medals, 3 Gold Australian Mathematical Olympiad certificates and 4 Australian Mathematics Competition medals! ... just to mention a few.

Pure genius indeed!

shaon0 · Jul 20, 2009

kurt.physics said:
Add to this a Silver medal at the 2008 IMO, a Gold and 2 Silver Asian Pacific Mathematical Olympiad medals, 3 Gold Australian Mathematical Olympiad certificates and 4 Australian Mathematics Competition medals! ... just to mention a few.

Pure genius indeed!

lol. You sound like a stalker.

kurt.physics · Jul 20, 2009

shaon0 said:
lol. You sound like a stalker.

lol What could possibly make you think that ;D

buchanan · Jul 21, 2009

Here is Terry Tao's lecture he gave at the IMO:

http://terrytao.files.wordpress.com/2009/07/primes.pdf

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