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.

数学竞赛几何理论基础：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.

数学竞赛几何理论基础：分角定理、张角定理，推理证明

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.

https://m.toutiaoimg.cn/a6816353938204262927/?app=news_article_lite&is_hit_share_recommend=0

IMO questions could not be good Math of deep meaning, given that the contestants have to solve the tricky problems in a short time frame of 2 to 3 hours…IMO Prize is just an indication of Math capability, we can’t equate IMO winners as Mathematicians.

【专访美国奥数队总教练：奥数比赛对一个国家的数学水平有用吗？】复制这条信息€80avm€a56OR2€后打开👉今日头条极速版👈

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?

【四部委发文加强数学研究、国际奥数赛中美夺冠，如何看这几天的“数学热”？】

https://m.toutiaocdn.com/group/6716317738618323467/?app=news_article_lite×tamp=1563792727&req_id=201907221852070100170311690747625&group_id=6716317738618323467&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(想看更多合你口味的内容，马上下载 今日头条)

http://app.toutiao.com/news_article/?utm_source=link

【打破美国35年纪录的天才，与中国奥数、信息学双料冠军的巅峰对决】

https://m.toutiaocdn.com/group/6714952524924715531/?app=news_article_lite×tamp=1563521442&req_id=20190719153042010152029169696E814&group_id=6714952524924715531&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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】

https://m.toutiaocdn.com/group/6715594110440309261/?app=news_article_lite×tamp=1563682066&req_id=20190721120745010025067136905FD11&group_id=6715594110440309261&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

**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】

https://m.toutiaocdn.com/group/6716112894641046024/?app=news_article_lite×tamp=1563733425&req_id=20190722022345010152042134236B8C3&group_id=6716112894641046024&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(想看更多合你口味的内容，马上下载 今日头条)

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【为什么中国奥数那么厉害，却出不了像高斯、欧拉这种级别的数学家？】

https://m.zjurl.cn/answer/6697035189161296136/?app=news_article&app_id=35&share_ansid=6697035189161296136&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(想看更多合你口味的内容，马上下载 今日头条)

http://app.toutiao.com/news_article/?utm_source=link

【如果让陈景润、华罗庚这种级别的数学家去参加高考，数学能答满分吗？为什么？】

https://m.zjurl.cn/answer/6643182628168007940/?iid=63376357991&app=news_article&share_ansid=6643182628168007940&app_id=35&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

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。

(想看更多合你口味的内容，马上下载 今日头条)

http://app.toutiao.com/news_article/?utm_source=link

**Problem A3**

A function f is defined on the positive integers by:

for all positive integers n,

Determine the number of positive integers n less than or equal to 1988 for which f(n) = n.

What is the explanation of the solution of problem 3 from IMO 1988? by Alon Amit

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.

**Reference**:

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

**Calculate**:

Let

Or:

Quite messy to expand out:

This 14-year-old vienamese student in Berlin – Huyen Nguyen Thi Minh discovered a smart trick using the identity:

or more general,

He multiplies x by (3-1):

.

.

.

**When n = 16,**

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:**

https://www.imo-official.org/results.aspx

**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

Prove:

Proof:

13 Odd digits = {3.14159265358979323846264 }

11 Even digits

Put 13 odds into 12 brackets, by Pigeonhole Principle, there is certainly one bracket where

2n multiplies with any number will always give even.

The product of 2n with the other 11 brackets will always be even.

Therefore

(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**:

Prove:

**Proof**:

Apply (3):

a³ + b³ = (a+b)**(a² -ab + b²) **≥ (a+b)**ab**

Note:

**a² -ab + b²**= (a-b)² + ab ≥ **ab**

since (a-b)² ≥ 0

Symmetrically,

Add 3 RHS:

[QED]

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).