中国大学考研 Concours Masters degree 数学

This was one of my weekly “Colle” (oral test ) question in 1st Year University 1978 Classe Préparatoire (Math Supérieure) = “Analysis” module.

Sum of Squares :

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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French Concours & 科举 (Chinese Imperial Exams)

French Concours (Entrance Exams for Grandes Écoles) was influenced by Chinese Imperial Exams (科举\ko-gu in ancient Chinese, today in Hokkien dialect) from 7th century (隋朝) till 1910 (清末).  The French Jesuits priests (天主教耶稣教会) in China during the 16th -18th centuries ‘imported’ them to France, and Napoléon adopted it for the newly established Grande École Concours (Entrance Exams), namely, “École Polytechnique” (a.k.a. X).

The “Bachelier” (or Baccalauréat from Latin-Arabic origin) is the Xiu-cai (秀才), only with this qualification can a person teach school kids.

With Licencié (Ju-ren 举人) a qualification to teach higher education.

Concours was admired in France as meritocratic and fair social system for poor peasants’ children to climb up the upper social strata — ” Just study hard to be the top Concours students”! As the old Chinese saying: “十年寒窗无人问, 一举成名天下知” (Unknown as a poor student in 10 years, overnight fame in whole China once top in Concours). Today,  even in France, the top Concours student in École Polytechnique has the honor to carry the Ensign (flag) and be the first person  to march-past at Champs-Elysées in the National Day Parade.

Concours has its drawback which, albeit having produced top scholars and mandarins, also created a different class of elites to oppress the people. It is blamed for rapidly bringing down the Chinese Civilization post-Industrial Age in the last 200 years. 5 years before the 1911 Revolution, the 2nd last Emperor (光绪) abolished the 1,300- year-old Concours but was too late. Chinese people overthrew the young boy Emperor Puyi (溥仪) to become a Republic from 1911.

A strange phenomenon in the1,300-year Concours in which only few of the thousands top scorers — especially the top 3 : 状元, 榜眼, 探花 e.g. (唐)王维, (北宋)苏东坡, 奸相(南宋)秦桧,贪污内阁首輔(明)严嵩… — left their names known in history, while those who failed the Concours were ‘eternally’ famous in Literatures (the top poets LiBai 李白 and DuFu 杜甫), Great writers (吴承恩, 曹雪芹, 蒲松龄, 罗贯中, 施耐庵), Medicine (《本草綱目》李时珍, 发明”银翹散”的吳鞠通), Taipeng Revolution leader (洪秀全)….

Same for France, not many top Concours students in X are as famous in history (except Henri Poincaré) as Evariste Galois who failed tragically in 2 consecutive years.

The French “grandiose ” in Science – led by Pascal, Fermat, Descartes, Fourier, Laplace, Galois, etc. — has been declining after the 19th century, relative to the USA and UK,  the Concours system could be the “culprit” to blame, because it has produced  a new class of French “Mandarins”  who lead France now in both private and government sectors. This Concours system opens door to the rich and their children, for the key to the door lies in the Prépas (Classes Préparatoires, 2-year post-high school preparatory classes for grandes écoles like X), where the best Prépas are mostly in Paris and big cities (Lyon, Toulouse…), admit only the top Baccalauréat (A-level) students. It is impossible for poor provinces to have good Prépas, let alone compete in Concours for the grandes écoles. The new elites are not necessary the best French talents, but are the privilegés of the Concours system who are now made leaders of the country.

Note: Similar education & social problem in Japan, the new Japanese ‘mandarins’ produced by the competitive University Entrance Exams (Todai 东大) are responsible for the Japanese post-Bubble depression for 3 decades till now.

These ‘Mandarins’ (官僚) of the past and modern days (Chinese, French, Korean “Yangban 양반 両班 “, Japanese) are made of the same ‘mould’ who think likewise in problem solving, protect their priviledged social class for themselves and their children, form a ‘club mafia’ to recruit and promote within their alumni, all at the expense of meritocracy and well-being of the corporations or government agencies. The victim organisation would not take long to rot at the roots, it is a matter of time to collapse by a sudden storm overnight — as seen by the demise of the Chinese Qing dynasty, the Korean Joseon dynasty (朝鲜李氏王朝), and the malaise of present French and Japanese economies.



Solution Ecole Polytechnique -Ecole Normales Superieures Concours 2013

We shall walk through the problem at little steps and day-by-day, not so much interest in the final solution per se , but with a higher aim to revise the modern algebra lessons along the way.

The French professors who designed this problem had done beautifully using all the concepts learned in the 2-year Classe Préparatoire (or Prépa, equivalent to Bachelor degree in Math & Science) – it is like an orchestra composer who pieces together all instruments to play a beautiful symphony – the catch is that the student must have a good grasp of all algebra topics.


I – Operators on the functions with finite support

1. a. Prove that V is a vector subspace of \mathbb{C} ^{\mathbb{Z}}.

Recall the definitions:

\mathbb{C} ^{\mathbb{Z}} = v.s.{f:\mathbb{Z} \mapsto \mathbb{C}}

Support = supp(ƒ) = {k \in \mathbb{Z} \mid f(k) \neq 0}

V = {f | supp(f) is a finite set}.

To prove V a vector subspace of \mathbb{C} ^{\mathbb{Z}},
1) V must be non-empty?

V contains null function, so not empty subset of \mathbb{C} ^{\mathbb{Z}}

2) closed under vector addition?

supp (f+g) \subset supp(f) \cup supp (g)

3) closed under scalar multiplication?

supp(\alpha f) = supp(f) for \alpha \neq 0

Given ƒ ∈ \mathbb{C} ^{\mathbb{Z}} , E(ƒ) ∈ \mathbb{C} ^{\mathbb{Z}}
by E(ƒ)(κ)= ƒ(κ+1), κ ∈ \mathbb{Z}

1. b. Show that E∈ L( \mathbb{C} ^{\mathbb{Z}} ) and V is stable by E.

E by definition is an operator of shift, hence a linear transformation, thus
E∈ L( \mathbb{C} ^{\mathbb{Z}} ).

[Solutions for XLC paper]:

Click to access Polytechnique_MP_A_XLC_corr_1.pdf

Translated Ecole Polytechnique & Ecole Normales Superieures Concours 2013

Math Paper A (XLC)
Duration: 4 hours

Use of calculator disallowed.

We propose to study the algebras of the remarkable endomorphisms of vector spaces of infinite dimension.


An i^{th} root of unity is called primitive if it generates the group of i^{th} roots of unity.

In this problem, all vector spaces are over the base field of complex numbers field \mathbb{C} .

If ε is a vector space, the algebra of the endomorphisms of ε is denoted by L(ε), and the group of the automorphisms of ε is denoted by GL(ε).

Id_{\varepsilon} denotes the identity mapping of ε.

If u ∈ L(ε), \mathbb{C}[u] denotes the sub-algebra \{P(u) \mid P \in \mathbb{C}[X] \} of L(ε) of the Polynomials in u.

\mathbb{C} ^{\mathbb{Z}} denotes the vector space of the functions of \mathbb{Z} to \mathbb{C} .

If ƒ is the function of \mathbb{Z} to \mathbb{C}, supp(ƒ) denotes the set of κ ∈ \mathbb{Z} such that ƒ(κ) ≠0
We call this set the support of ƒ.

Throughout the problem, V denotes the set of functions of \mathbb{Z} to \mathbb{C} of which the support is a finite set.

I – Operators on the functions with finite support

1. a. Prove that V is a vector subspace of \mathbb{C} ^{\mathbb{Z}}.
Given ƒ ∈ \mathbb{C} ^{\mathbb{Z}} ,
we define E(ƒ) ∈ \mathbb{C} ^{\mathbb{Z}}
E(ƒ)(κ)= ƒ(κ+1), κ ∈ \mathbb{Z}

1. b. Show that E∈ L( \mathbb{C} ^{\mathbb{Z}} ) and V is stable by E.

In the following, E denotes uniquely the endomophism of V induced.

2. Show that E ∈ GL(V).

3. For i \in  \mathbb{Z} , we define v_i  \in  \mathbb{C} ^{\mathbb{Z} } by:

v_i(k) = \begin{cases} 1 , & \text{if } k = i\\ 0, & \text{if } k \neq i \end{cases}

3.a. Prove that the family \{v_i\}_{ i \in \mathbb{Z}} is the base of V.

3.b. Calculate E(v_i ).
Let \lambda, \mu \in \mathbb{C} ^{\mathbb{Z}} ,
we define the respective linear mappings F, H \in L(V)
by: H(v_i) = \lambda(i)v_i
and F(v_i) = \mu(i)v_{i+1},   i \in \mathbb{Z}

4. Prove that
H  \circ E = E \circ H + 2E
if and only if for all i \in \mathbb{Z}, \lambda(i) = \lambda(0)-2i

In the remaining of Section I (but not in the following Sections), we asume the conditions in question 4 are verified.

5. Prove that
E \circ F = F \circ E + H
if and only if \forall i \in \mathbb{Z},  \mu(i) = \mu(0) +I(\lambda(0) -1) -i^2

6.a. Prove that for f \in V the vector space generated by H^n(f), n\in \mathbb{N} has finite dimension.

6.b. Deduce that a vector subspace non-reduced to {0} of V, stable by H, contains at least one of the v_i.

In the remaining of Section I (but not in the following Sections), we asume the conditions in question 5 are verified and
\lambda(0)=0, \mu(0)=1

7.a. Prove that F \in GL(V)

7.b. Prove that E and F are not of finite order in the group GL(V).

7.c. Calculate the kernel of H and prove that H^r \neq Id_{v} \text{ for } r \geq 1

8. \mathbb{C}[X] denotes the polynomials with complex number coefficients in one indeterminate X.

8.a. Prove that \mathbb{C}[E] is isomorphic (as Algebra) to \mathbb{C}[X].

8.b. Prove that \mathbb{C}[F] is isomorphic (as Algebra) to \mathbb{C}[X].

8.c. Prove that \mathbb{C}[H] is isomorphic (as Algebra) to \mathbb{C}[X].

II – Interlude

In all the rest of the problem, we fix an odd interger \ell \geq 3 and q a \ell^{th} primitive root of unity.

9. Prove that q^2 is a \ell^{th} primitive root of unity.

Let W_{\ell}  = \displaystyle \bigoplus \limits_ {0 \leq i  < \ell } \mathbb{C} v_i \text{ and } a \in \mathbb{C}^{*}

10. Consider the element G_a \text { of  } L(W_{\ell}) \text { of which the matrix with base }  \{v_i\}_{0 \leq i  < \ell } is :

\begin{pmatrix} 0 & 0 & 0 & \ldots & 0 & a\\ 1 & 0 & 0 & \ldots & 0 & 0\\ 0 & 1 & 0 & \ldots & 0 & 0\\ \vdots & 0 &  \ddots & \ddots &  \vdots & \vdots\\  0 & \vdots &  \ddots & \ddots &  0& 0\\ 0 & 0 &\ldots & 0 & 1 & 0 \end{pmatrix}

10.a. Calculate {G_a}^{\ell}.
Prove that G_a \text{  is diagonalisable.}

10.b. Let b \ell^{th} root of a.
Calacute the eigenvectors of G_a and the associated eigenvalues in function of b, q and v_i .

Let’s define a linear mapping P_a : V \to V by
P_a(v_i) =a^{p}v_r \text{ where } i \in \mathbb{Z} , and
define r and p respectively the residue and the quotient of the euclidian division of i by ℓ; ie:
i = p\ell + r
0 \leq r < \ell , p \in \mathbb{Z}

11. Prove that P_a is a projector of image W_\ell.

III – Quantum Operators

12. Prove that
H  \circ E = q^{2}E \circ H if and only if
\forall  i \in \mathbb{Z}, \lambda(i) = \lambda(0) q^ {-2i}

In the following problem, we asume the conditions in question 12 are verified and
\lambda (0) \neq 0

13. Prove that H \in GL(V).

14. Prove that
E \circ F = F \circ  E  + H - H^{-1} if and only if
\forall  i \in \mathbb{Z},  \mu(i) = \mu(i-1) + \lambda(0)q^ {-2i} -\lambda(0)^{-1}q^{2i}

In the following problem, we asume the conditions in question 14 are verified.

15.a. Prove that \lambda \text{ and } \mu are periodic over \mathbb{Z}, of periods dividing \ell.

15.b. Prove that the period of \lambda \text { is equal to } \ell.

15.c. Prove that the period of \mu is also equal to \ell.

16. Let C = (q -q^{-1)}E \circ F+q^{-1}H +qH^{-1} with H^{-1} being inverse of H.

16.a. Prove that
C = (q -q^{-1)}F \circ E + qH +q^{-1}H^{-1}.

16.b. For i \in \mathbb{Z} , prove that v_i is an eigenvector of C.

16.c. Deduce that C is a homothety of v of which we calculate the ratio of R(\lambda(0), \mu(0),q)   in function of \lambda(0), \mu(0), q.

16.d. Let’s fix q \text{ and } \lambda(0). Prove that the mapping
\mu(0) \mapsto R(\lambda(0), \mu(0),q) is a bijection of \mathbb{C} onto \mathbb{C}.

16.e. Let’s fix q \text{ and } \mu(0) . Prove that the mapping
\lambda(0) \mapsto  R(\lambda(0), \mu(0),q) is a surjection of \mathbb{C}^{*} onto \mathbb{C} but not a bijection.

IV – Modular Quantum Operators

Let \ell, W_\ell, a, P_a like in the Section II. We say an element \phi of L(V) is compatible with P_a if
P_a \circ \phi \circ P_a = P_a \circ \phi

17.a. Prove that if \phi \in L(V) is commutative with P_a ,then \phi is compatible with P_a.

17.b. Prove that H \text{ and } H^{-1} are compatible with P_a.

Let U_q the set of endomorphisms \phi \in L(V) which are compatible with P_a.

18. Prove that U_q is a sub-algebra of L(V).

19. Prove that E \in U_q \text { and } F \in U_q.

20.a. Show that there exists an unique morphism of algebras \Psi_{a}: U_q \to L(W_{\ell}) such that:
\forall \phi \in U_q , \Psi_{a}(\phi) \circ P_a = P_a \circ \phi

20.b. Prove that \phi \in U_q is contained in the kernel of \Psi_{a} if and only if the image of \phi is a vector subspace of v generated by the vectors v_i - a^{p}v_{r} \text{ , } i \in \mathbb{Z} where i =p\ell + r is the euclidian division of i \text { by }  \ell.

21. Let’s study in this question \Psi_a(E).

21.a. Determine \Psi_a(E)(v_0).

21.b. Deduce \Psi_a(E^\ell).

21.c. Calculate the dimension of the vector subspace of \mathbb{C} [\Psi_a(E)]

21.d. Calculate the eigenvectors of \Psi_a(E)

22. Let W a non zero sub-space of W_\ell stable by \Psi_a(H).

22.a. Show that W contains at least one of the vectors v_i.

22.b. What do you say if W is in addition stable by \Psi_a(E) ?

23. Give a necessary and sufficient condition on R(\lambda(0), \mu(0),q) in order for the operator \Psi_a(F) to be nilpotent.



Ecole Polytechnique Concours 2013

French Math Exams paper is called “Composition”, it is unlike English Math paper solving different independent questions. In fact “Composition” is made up of many inter-dependent smaller questions, they together step-by-step lead to proving some Math theorems or corollary.

Before every Math composition, the French Math professor would tell the students the test scope covers all Math they learn thus far from primary school till today. Quite similar to sitting for any English language Composition, the scope of  vocaburary and grammar covers everything we learn since day 1 in primary school. Math is, after all, a “language” of science and logic.

Look at this year Ecole Polytchnique (and Ecole Normales Supérieures) Math Composition below:

[Note: My next few blogs will contain the English translation, and hopefully the solution contributed from the comments by blog readers. ]

It is notoriously famous for being very tough. It needs 2 years of preparation  after Baccalaureat (A-level) in the Classes Préparatoires (Maths Supérieures, Maths Spéciales), located not in universities but in few prestigious ancient Lycées (High Schools) selected by Napoléon eg. Lycée Louis Le Grand (Paris), Lycée Henri IV, Lycée Pierre de Fermat (Toulous), Lycée Du Park (Lyon)… taking in only the brightest Baccalaureat high-school students in Math (only 7.5% of each year High-school cohorts from Baccalaureat).

200 years ago the 19th century Math genius (father of Group Theory and Modern Algebra) Evariste Galois failed this Ecole Polytechnique “Composition” Exams twice because he was too good for the Examiners to understand him. The inventor of Topology Henri Poincaré topped in this Exams, while Charles Hermite (Galois’s 15 years junior from the same professor Richard of Lycée Louis Le Grand) was in the last position, almost failed!

Note: Ecole Normales Supérieures and Ecole Polytechnique combine their Concours entrance exams together in recent years.

科举 -Concours -Scholars : Exams Origin

Very good and brief summary of ancient Chinese Exams system (科举 \keju in Mandarin or \Kor-cou in Fujian 福建 where 16th century French Jesuit missionaries stationed in coastal China), copied by French competitive “Concours” (Entrance exams to elite universities Grandes Écoles) in 18th century, England’s Civil Servant Exams, Singapore Government elite scholars in Admins Services.

The Exams system has good and bad influences. Good being it is democratic and meritocratic, bad being prone to cheating.

Ancient Chinese Examination System Made Relevant: