# 《数学与人类文明》Mathematics & Civilizations

《数学与人类文明》数学与现代文明：

1. 哲学：希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴， 达芬奇, Golden Ratio

3. 工业革命：Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .

https://m.toutiaoimg.cn/group/6919012894134764046/?app=news_article&timestamp=1611057637&group_id=6919012894134764046&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# IMO 2020 Solutions

IMO 2020 : China World Team Champion ( 5 Golds + 1 Silver), the only one in the World with Perfect Score.

This Chinese IMO coach comments :

IMO 2020 is the easiest in the past 10 years, compared to 2015 (tougher) & 2018 (toughest).

https://m.toutiaoimg.cn/a6881188536880464384/?app=news_article&is_hit_share_recommend=0&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Math Olympiad is harmful

The harms to kids math education:
1. Teach too early the higher math, only “acrobatic technique” , not genuine math education.
2. Waste parents’ money in unnecessary tuition for Olympiad math.
3. Pressure on kids.

# 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

# Inequalities

Inequalities (for IMO Math) : Just remember this diagram (to reconstruct from memory each time)

AM : Arithmetic Mean
GM : Geometric Mean # 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. # IMO USA Coach’s Advices

【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?

# Chinese Math Olympiad (Primary) Trick vs Singapore Modeling Math

Both Methods (I) & (II) below does not use Algebra, which is not taught to primary school kids until secondary school (above 12 or 13 years old).

(I) Ancient Chinese Arithmetic Method:

When the total of their ages is 40, how old will they be? 【这位妈妈太绝了！竟把小学6年奥数化为13句口诀】

Trick: Age difference does not change over time, they + or – in tandem. When 2 ages change, the multiple between them also changes. https://m.toutiaocdn.cn/group/6636154201682477582/?iid=65171531104&app=news_article_lite&timestamp=1552573996&group_id=6636154201682477582&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

(II) Singapore Modeling Math:

［例2］ Compare the Ancient Chinese Arithmetic (I) & Singapore Modeling Math (II):

• (II) is better for young kids to understand visually the logic behind,
• (I) is more a memorised trick but poor math education pedagogy. In [例 2] it is very hard for kids to understand why the 2 steps (40+4) & (40 – 4), although they lead to the correct answer.

Notes:

https://tomcircle.wordpress.com/2019/03/14/chinese-math-olympiad-primary-trick/

# RMM Olympiad 2019   https://m.toutiaocdn.com/i6662602678880698894/?iid=63376357991&app=news_article_lite&timestamp=1551357660&group_id=6662602678880698894&tt_from=android_share&utm_medium=toutiao_android&utm_campaign=client_share

# Inequalities : RMS, AM, GM, HM   # Fields Medals 2018 Picture Order (From Left) :1. 2. 3. 4.

1. Caucher Birkar (UK / Kurdish – Iran, 40)

https://www.bbc.com/news/science-environment-45032422

2. Alessio Figalli (Italy, 34)
https://www.quantamagazine.org/alessio-figalli-a-mathematician-on-the-move-wins-fields-medal-20180801/

3. Akshay Venkatesh (Australia / India, 36)

Studies number theory and representation theory.

https://www.quantamagazine.org/fields-medalist-akshay-venkatesh-bridges-math-and-time-20180801/

4. Peter Scholze (Germany, 30)
Intersection between number theory and geometry # Quora : How likely is it that a mathematics student can’t solve IMO problems?

How likely is it that a mathematics student can’t solve IMO problems?

Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh

https://www.quora.com/How-likely-is-it-that-a-mathematics-student-cant-solve-IMO-problems-Is-there-a-fear-of-embarrassment-in-being-a-math-Ph-D-who-cant-solve-problems-that-high-school-students-can/answer/Cornelius-Goh?share=311c5a88&srid=oZzP

# 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

https://www.quora.com/What-is-the-explanation-of-the-solution-of-problem-3-from-IMO-1988/answer/Alon-Amit?share=7719956f&srid=oZzP

# Math Olympiad (Primary Schools) 一筐鸡蛋

One basket of eggs.
1粒1粒拿，正好拿完。
Remove 1 by 1, nothing left in basket.

2粒2粒拿，还剩1粒。
Remove 2 by 2, one left in basket.

3粒3粒拿，正好拿完。
Remove 3 by 3, nothing left in basket.

4粒4粒拿，还剩1粒。
Remove 4 by 4, one left in basket.

5粒5粒拿，还差1粒才能拿完。
Remove 5 by 5, short of one to complete.

6粒6粒拿，还剩3粒。
Remove 6 by 6, 3 left in basket.

7粒7粒拿，正好拿完。
Remove 7 by 7, nothing left in basket.

8粒8粒拿，还剩1粒。
Remove 8 by 8, one left in basket.

9粒9粒拿，正好拿完。
Remove 9 by 9, nothing left in basket.

At least how many eggs are there in the basket?

[Hint] This is a Chinese Remainder Problem (” 韩信点兵”)

—— [Solution] —–

Let there be minimum X eggs in the basket.

Remove 1 by 1, nothing left in basket:
X = 0 mod (1) …
=> trivial & useless !

Remove 2 by 2, one left in basket:
X = 1 mod (2) … 

Remove 3 by 3, nothing left in basket:
X = 0 mod (3) … 

Remove 4 by 4, one left in basket:
X = 1 mod (4) … 

Remove 5 by 5, short of one to complete:
X = -1 mod (5) … 

Remove 6 by 6, 3 left in basket:
X = 3 mod (6) … 

Remove 7 by 7, nothing left in basket:
X = 0 mod (7) … 

Remove 8 by 8, one left in basket:
X = 1 mod (8) … 

Remove 9 by 9, nothing left in basket:
X = 0 mod (9) … 

Simplify  = 
X = 3 mod (6)
X = 3 mod (3×2)
X = 3 mod (3)
X = 0 mod (3) … 

Notice , ,: X is multiple of 3, 7, 9
=> X is mulitple of LCM (3,7,9) = 63
X = 0 mod (63) … [3,7,9]

Similarly,
X = 1 mod (2)
X = 1 mod (4)
X = 1 mod (8)
=> X = 1 mod (LCM {2, 4, 8}) [Note @]
=> X = 1 mod (8) … 

[Note @]: $\forall n,m,t,t',t" \in \mathbb{ N }$
X = 1 mod (2)
=> X – 1 = 2n
Similarly,
X = 1 mod (4) <=> X – 1 = 4m
X = 1 mod (8) <=> X – 1 = 8t = 4.(2n)t’ = 2.(4m)t”
These 3 equations <=> X = 1 mod (8)

Simplify the 9 equations by considering only the remaining 3 equations:

X = 0 mod (63) … [3,7,9]
X = -1 mod (5) … 
X = 1 mod (8) … 

Since (8, 5, 63 ) are co-primes pair-wise – ie (8, 5), (8, 63), (5, 63) are relative primes to each other in each pair – we can apply the Chinese Remainder Theorem to the last 3 equations: $\boxed {X = (63 \times 5).u1 +(63 \times 8).u2 +(5 \times 8).u3 \mod (8 \times 5 \times 63)}$ …[#]

By sequentially “mod” the equation [#] by 8, 5, 63, we get:

(63×5).u1 = 1 mod (8) … 
315.u1 = (39×8+3).u1 = 1 mod (8)
3.u1 = 1 mod (8)
=> u1 = 3 [as 9 = 1 mod (8)]

(63×8).u2 = -1 mod (5)… 
504.u2 = (50×11 – 1).u2 = -1 mod (5)
-1.u2= -1 mod (5)
u2 = 1 [as -1 = -1 mod (5)]

(5×8).u3 = 0 mod (63)… [3,7,9]
40.u3 = 0 mod (63)
u3 = 0 [as 0 = 0 mod (63) ]

[#]:
X = (63×5).u1 + (63×8).u2 +(5×8).u3 mod (8x5x63)
X = (63×5).3 + (63×8).1 + (5×8).0 mod (2520)

X = 63 x 23 = 1449 mod (2550) $\boxed {\text{Minimum Eggs } = 1449}$

# 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
Canada 9th
Singapore 10th [2012 Individual World’s Champion ]
Ukraine 11th
Thailand 12th
Romania 13th
France 14th

United Kingdom 22th

# Math Olympiad (kindergarten ?) Answer: (don’t scroll down until you have tried)

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Answer : 2889 = 5 