# 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.

https://m.toutiaoimg.cn/a6910816390312100360/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# 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.

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

# SS Chern 陈省身 on IMO Math

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?

# IMO 2019

USA Team (2016 & 2018 World Champion) :

China IMO Team :

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.

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# 大数学家会解 IMO /高考 / Concours吗 ？

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。

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# Quora: IMO 1988 Question 3

Problem A3

A function f is defined on the positive integers by:

for all positive integers n,

$f(1) = 1$
$f(3) = 3$
$f(2n) = f(n)$
$f(4n + 1) = 2f(2n + 1) - f(n)$
$f(4n + 3) = 3f(2n + 1) - 2f(n)$

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

# The Legend of Question Six (IMO 1988)

This question was submitted  by West Germany to the IMO Committee, the examiners could not solve it in 6 hours.

In the IMO (1988) only 11 contestants solved it,  one of them proved it elegantly. Terence Tao (13, Australia) only got 1 mark out of 7 in this question.

Solution

[INMO 1993]

# 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.

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

# Smart Algebraic Technique

Calculate:
$(3+1). (3^2 +1). (3^4 + 1)(3^8 +1).... (3^{32} +1)$

Let
$x = (3+1). (3^2 +1). (3^4 + 1)(3^8 +1).... (3^{2n} +1)$

Or:
$\displaystyle x = \sum_{n=0}^{n}(3^{2n} + 1)$

Quite messy to expand out:

$\displaystyle { \sum_{n=0}^{n} (3^{2n}) + \sum_{n=0}^{n}(1) = .... }$

This 14-year-old vienamese student in Berlin – Huyen Nguyen Thi Minh discovered a smart trick using the identity:
$\displaystyle { (a -1).(a + 1) = a^{2} - 1}$
or more general,
$\displaystyle \boxed { (a^{n} -1).(a^{n} + 1) = a^{2n} - 1 }$

He multiplies x by (3-1):

$x. (3-1) = (3-1)(3+1). (3^{2} +1)... (3^{2n} + 1)$
$2x = (3^{2} -1). (3^{2} +1)...(3^{2n} + 1)$

$2x = (3^{4} -1).(3^{4} +1) ... (3^{2n} + 1)$
.
.
.

$2x = (3^{4n} -1)$

$\displaystyle \boxed { x = (3^{4n} -1) / 2 }$
When n = 16,
$\displaystyle { x = (3^{64} -1) / 2 }$

$x = 1,716,841,910,146,256,242,328,924,544,640$

# 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:
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
Singapore 10th [2012 Individual World’s Champion ]
Ukraine 11th
Thailand 12th
Romania 13th
France 14th

United Kingdom 22th

# IMO Number Theory

IMO中的数论

Theorem:

U and V co-prime if there exist intergers m, n such that

m.U + n. V = 1

# Pigeonhole Principle

$\pi = 3.14159265358979323846264$
$\text{Let } a_1, a_2,\dots a_{24} \text{ represent the first 24 digits of } \pi$
Prove:
$(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text{ is even}$

Proof:

13 Odd digits = {3.14159265358979323846264 }

11 Even digits

$\text {12 brackets :}(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24})$

Put 13 odds into 12 brackets, by Pigeonhole Principle, there is certainly one bracket where
$(a_j - a_k) \text{ is a difference of 2 odds, which is an even = 2n}$

2n multiplies with any number will always give even.
The product of 2n with the other 11 brackets will always be even.

Therefore
$(a_1 - a_2)(a_3 - a_4) \dots (a_{23} - a_{24}) \text { is even}$

# 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:
Prove:
$\frac{1}{a^{3}+b^{3}+abc} + \frac{1}{b^{3}+c^{3}+abc} + \frac{1}{c^{3}+a^{3}+abc} \leq \frac{1}{abc}$

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

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

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

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

$\frac{a+b+c}{a+b+c} = 1$

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

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

[QED]

# 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).