# 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,
$\displaystyle { x = (3^{64} -1) / 2 }$

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