IMO Geometry Techniques 几何理论基础:分角定理、张角定理,推理证明

IMO Math usually contains 1 or 2 Geometry questions.

France, UK, Singapore, and some countries which reduce Secondary school syllabus in Euclidien Geometry, are disadvantaged in scoring Gold.


IMO (1988) 6th Problem

In the 1988 IMO only 11 contestants solved this 6th problem, including 2 future Fields Medalists : Terrence Tao (12 years old) & G. PERLMAN.

The most elegant solution came from the 17 year-old Balgarian contestant using “Reductio Absurdum” Proof : Simple & “Violent” way.

IMO USA Coach’s Advices


2019 both China and USA co-win the IMO Team Champion, both teams consist of almost Chinese ethnic students (except 1 white american) & Chinese coaches.

Key Points :

IMO questions : exclude Calculus.

IMO Boot Camp: 3 month-training.

Calculus : In High Schools just learn formula & apply, in university learn the theory.

France is a Math power but weak in IMO, why?



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IMO 2019


USA Team (2016 & 2018 World Champion) :

China IMO Team :

【2019国际数学奥赛结果公布 中美两国并列团体冠军】

Two observations:
1. Why almost all Chinese in top China, USA, Canada, NZ, Singapore teams?
Either Chinese chase the wrong Math (IMO) education, or the western Math Power countries (UK, France, Germany…) are RIGHT to ignore IMO education?
2. Singapore team skewed heavily in 2 schools: RI (5) + HCI (1). Also 0 girl.

【解读中美数学奥赛并列第一,敬请期待对话美国队总教练 | 袁岚峰】

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大数学家会解 IMO /高考 / Concours吗 ?


答案是: 不能!

19 CE Evariste Galois 是 ”抽象代数” 群论之父 (Abstract Algebra – Group Theory), 大学入学考试 (Concours = 法国科举) 连续2年不及格 – 因为他准备不充足,不适应考题的技巧。

反之,中国IMO 奥数2届满分金牌的一位学生,北大数学系不能毕业

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Prof ST Yau’s 丘成桐 Talk to Chinese Youth on Math Education 

Prof ST Yau 丘成桐 , Chinese/HK Harvard Math Dean, is the only 2 Mathematicians in history (the other person is Prof Pierre Deligne of Belgium) who won ALL 3 top math prizes: Fields Medal 1982 (at 27, proving Calabi Conjecture), Crafoord Prize (1994) , Wolf Prize (2010).

Key Takeaways :

1. On Math Education
◇ Compulsary Math training for reasoning skill applicable in Economy, Law, Medicine, etc.
◇ Study Math Tip: read the new topic notes 1 day before the lecture, then after lecture do the problems to enhance understanding.
◇ Read Math topics even though you do not understand in first round, re-read few more times,  then few days / months / years / decades later you will digest them. (做学问的程序).
◇ Do not consult students in WHAT to teach, because they don’t know what to learn.
◇ Love of Math beauty is the “pull-factor” for motivating  students’ interest in Math.
◇ Parental Pressure.

2. “3D” facial photo using Math

3. Pi-Music: 1 = “do”, 2 = “re”, 3 =”me”…
Pi =3.1415926…

4. Math Olympiad: Prof ST Yau had criticised publicly it as a bad Math training, not the “real” Math. 

An audience tested Prof ST Yau on a Math (Accounting) Puzzle which he couldn’t  solve on the spot. He said Mathematicians are poor in +-×÷ arithmetic. 

5. Chinese students in USA: China sends over 200,000 students to USA universities. They are good in secondary / high school Math with known solutions,  but poor in graduate PhD Math which requires “out-of-the-box” independent thinking skill for finding unknown solutions. Recent few years Chinese students (eg. Stanford Prof 李骏 : 1989 Harvard PhD)  in USA have improved standard in PhD research.

6. Research is not for fame. It takes many years to think through an interesting topic.


1. Prof ST Yau’s Best Seller Book 《The Shape of Inner Space》avail @ NLB (Ref #530.1) 11 copies in most NLB branches@ AMK, Bishan etc.

2. Interview Prof ST Yau by HK TV (Cantonese)

3. 丘成桐 (2008) 评中国 和 美 国的教育 : 中国学生不爱看课外书, 因为考试太重, 课余时间花在玩电脑游戏。

4.  丘成桐 (2016): 中国大学本科要注重基础教育, 才能培养世界级一流人才

IMO 2015 USA beat China after 20 Years


The result is not surprising to China but to USA:
♢Recently China government bans IMO training in schools.
♢Obama was surprised that the USA IMO team consists of predominantly Chinese American students.

IMO Math is like ‘Acrobatics’ to real ‘Kung-fu’, it is not real Math education, but special ‘cute’ techniques to solve tough ‘known’ solution problems. Real Math is long R&D solving problems with UNKNOWN solution (eg. Fermat’s Last Theorem, Riemann Conjecture,…)

2 types of Math: Algorithmic or Deductive (演绎). Chinese long traditional ‘abacus’ mindset, procedural computational Math is Algorithmic, applied to special cases (eg. astronomy, calendar, agriculture, architecture, commerce,…). European Greek’s Euclid deductive, step-by-step axiom-based proofing, is theoretical, generalized in all cases (Geometry, Abstract Algebra,…)

Look at the Fields Medal (aka ‘Nobel Prize’ of Math) super-power – France – which has produced 1/3 of the Fields Medalists, but performing so-so in IMO. In contrast, China has ZERO Fields Medalists, albeit dominating IMO championship for more than 2 decades!

IMO 2015:
USA 1st,
China 2nd,
South Korea 3th,
North Korea 4th,
Vietnam 5th,
Australia 6th
Iran 7th
Russia 8th
Canada 9th
Singapore 10th [2012 Individual World’s Champion ]
Ukraine 11th
Thailand 12th
Romania 13th
France 14th

United Kingdom 22th


IMO Technique

(a+b)³ = a³ + 3a²b+ 3ab² + b³

Different equivalent forms:
(1):(a+b)³ = a³ + b³+3ab(a+b)
(2):a³ + b³ = (a+b)³ – 3ab(a+b)
(3): a³ + b³ = (a+b)(a² -ab + b²)
(4):(a+b)³ – ( a³ + b³ ) = 3ab(a+b)

1997 USAMO Q5:
\frac{1}{a^{3}+b^{3}+abc}   +   \frac{1}{b^{3}+c^{3}+abc}   + \frac{1}{c^{3}+a^{3}+abc}   \leq  \frac{1}{abc}

Apply (3):
a³ + b³ = (a+b)(a² -ab + b²) ≥ (a+b)ab

a² -ab + b²= (a-b)² + ab ≥ ab
since (a-b)² ≥ 0

\frac{abc}{a^{3}+b^{3}+abc}       \leq \frac{abc}{(a+b)ab + abc}  = \frac{c}{a+b+c}

\frac{abc}{b^{3}+c^{3}+abc}       \leq \frac{a}{a+b+c}

\frac{abc}{c^{3}+a^{3}+abc}       \leq \frac{b}{a+b+c}

Add 3 RHS:
\frac{a+b+c}{a+b+c} = 1

\frac{abc}{a^{3}+b^{3}+abc}   +   \frac{abc}{b^{3}+c^{3}+abc}   + \frac{abc}{c^{3}+a^{3}+abc}   \leq 1

\frac{1}{a^{3}+b^{3}+abc}   +   \frac{1}{b^{3}+c^{3}+abc}   + \frac{1}{c^{3}+a^{3}+abc}   \leq  \frac{1}{abc}


IMO Super-Coach: Rukshin

Rukshin at 15 was a troubled russian kid with drink and violence, then a miracle happened: He fell in love with Math and turned all his creative, aggressive, and competitive energies toward it.

He tried to compete in Math olympiads, but outmatched by peers. Still he believed he knew how to win; he just could not do it himself.

He formed a team of schoolchildren a year younger than he and trained them.
At 19 he became an IMO coach who produced Perelman (Gold IMO & Fields/Clay Poincare Conjecture). In the decades since, his students took 70 IMO, include > 40 Golds.

Rukshin’s thoughts on IMO:

1. IMO is more like a sport. It has its coaches, clubs, practice sessions, competitions.

2. Natural ability is necessary but NOT sufficient for success: The talented kid needs to have the right coach, the right team, the right kind of family support, and, most important, the WILL to win.

3. At the beginning, it is nearly impossible to tell the difference between future (Math) stars and those who will be good (at IMO) but never great (Mathematician).