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Key Facts: Concours X-ENS-ESPCI Exam

6 institutions

École Polytechnique, the four ENS (Ulm, Lyon, Paris-Saclay, Rennes) and ESPCI Paris

polytechnique.edu

13–17 April 2026

Written papers of the common X-ENS-ESPCI bank (MP, MPI, PC, PSI)

Calendrier CPGE Polytechnique 2026

27 May 2026

Polytechnique admissibilité results; PT candidates sit the Banque PT instead

polytechnique.edu

220 €

SCEI fee for Polytechnique and for ESPCI in PC; the four ENS are free, boursiers exempt

SCEI 2026

24 July 2026

Admission results communicated individually from 14:00

polytechnique.edu

Separate juries

Each school sets its own coefficients, admissibility thresholds and orals

polytechnique.edu / ens.psl.eu

France's most prestigious and mathematically demanding scientific Grandes Écoles competitive entrance examination, feeding École Polytechnique (l'X), the four Écoles Normales Supérieures (Ulm, Lyon, Paris-Saclay, Rennes), and ESPCI Paris across the MP, MPI, PC, and PSI preparatory streams.

Sample Concours X-ENS-ESPCI Practice Questions

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1Let G be a finite group of order |G| = 100 = 2^2 × 5^2. According to the Sylow theorems, what is the number n_5 of Sylow 5-subgroups of G, and what does this imply about the structure of G?
A.n_5 = 1, meaning G possesses a unique normal Sylow 5-subgroup of order 25.
B.n_5 = 2, meaning G is isomorphic to the direct product Z/4Z × Z/25Z.
C.n_5 = 4, meaning G has four conjugate Sylow 5-subgroups and cannot be abelian.
D.n_5 = 5, meaning the Sylow 5-subgroups partition the non-identity elements of G.
Explanation: By the Sylow theorems, the number of Sylow 5-subgroups n_5 must satisfy n_5 ≡ 1 (mod 5) and n_5 must divide |G|/5^2 = 4. The divisors of 4 are 1, 2, and 4. The only divisor satisfying n_5 ≡ 1 (mod 5) is n_5 = 1. Since any Sylow p-subgroup is unique if and only if it is normal, G necessarily contains a unique normal Sylow 5-subgroup of order 25.
2Let A ∈ M_3(R) be a real 3 × 3 matrix with characteristic polynomial χ_A(X) = (X - 2)^2 (X + 1). If dim(ker(A - 2 I_3)) = 1, what is the minimal polynomial μ_A(X) of A?
A.μ_A(X) = (X - 2)(X + 1)
B.μ_A(X) = (X - 2)^2 (X + 1)
C.μ_A(X) = (X - 2)(X + 1)^2
D.μ_A(X) = (X - 2)^3
Explanation: The minimal polynomial μ_A(X) shares the same irreducible factors as the characteristic polynomial χ_A(X), so μ_A(X) must be either (X - 2)(X + 1) or (X - 2)^2 (X + 1). The matrix is diagonalizable over C if and only if its minimal polynomial is split with simple roots. Since the geometric multiplicity dim(ker(A - 2 I_3)) = 1 is strictly less than the algebraic multiplicity 2, the eigenspace corresponding to eigenvalue 2 is not full, meaning A is not diagonalizable. Therefore, the multiplicity of the root (X - 2) in μ_A(X) must be 2, yielding μ_A(X) = (X - 2)^2 (X + 1).
3Consider the polynomial P(X) = X^4 + 1. Over which of the following fields is P(X) irreducible?
A.Over the field of real numbers R
B.Over the finite field F_3 = Z/3Z
C.Over the field of rational numbers Q
D.Over the finite field F_2 = Z/2Z
Explanation: Over Q, P(X) = X^4 + 1 is the 8th cyclotomic polynomial Φ_8(X), which is irreducible. Alternatively, substituting X = Y + 1 yields P(Y + 1) = Y^4 + 4Y^3 + 6Y^2 + 4Y + 2; applying Eisenstein's criterion with the prime p = 2 (which divides 4, 6, 4, 2 but 2^2 = 4 does not divide 2) proves irreducibility over Q. Over R, P(X) = (X^2 - X√2 + 1)(X^2 + X√2 + 1) factors into real quadratics. Over F_3, X^4 + 1 = (X^2 + X + 2)(X^2 + 2X + 2). Over F_2, X^4 + 1 = (X + 1)^4.
4In the vector space E = R_2[X] of real polynomials of degree at most 2, consider the evaluation linear forms f_0(P) = P(-1), f_1(P) = P(0), and f_2(P) = P(1). The family B* = (f_0, f_1, f_2) forms a basis of the dual space E*. What is the polynomial P_1(X) belonging to the dual basis (biorthogonal basis) B = (P_0, P_1, P_2) of E such that f_i(P_j) = δ_{ij}?
A.P_1(X) = (1/2)(X^2 - X)
B.P_1(X) = (1/2)(X^2 + X)
C.P_1(X) = X^2 - 1
D.P_1(X) = 1 - X^2
Explanation: The basis polynomials P_j are the Lagrange interpolation polynomials for nodes x_0 = -1, x_1 = 0, x_2 = 1. For j = 1, P_1(X) must satisfy P_1(-1) = 0, P_1(0) = 1, and P_1(1) = 0. The roots are -1 and 1, so P_1(X) = c(X - (-1))(X - 1) = c(X^2 - 1). Evaluating at X = 0 gives P_1(0) = -c = 1, hence c = -1 and P_1(X) = 1 - X^2.
5Let E be a finite-dimensional complex Hilbert space with inner product ⟨·, ·⟩, and let u ∈ L(E) be a self-adjoint operator with eigenvalues λ_1 ≤ λ_2 ≤ ... ≤ λ_n. According to the Courant-Fischer min-max theorem, how is the k-th eigenvalue λ_k characterized?
A.λ_k = min_{dim(V) = k} max_{x ∈ V, x ≠ 0} (⟨u(x), x⟩ / ⟨x, x⟩)
B.λ_k = max_{dim(V) = k} max_{x ∈ V, x ≠ 0} (⟨u(x), x⟩ / ⟨x, x⟩)
C.λ_k = min_{dim(V) = n - k} min_{x ∈ V, x ≠ 0} (⟨u(x), x⟩ / ⟨x, x⟩)
D.λ_k = (1/k) ∑_{j=1}^k max_{x ⊥ v_j} ⟨u(x), x⟩
Explanation: The Courant-Fischer min-max theorem states that for a self-adjoint operator with ordered eigenvalues λ_1 ≤ λ_2 ≤ ... ≤ λ_n, the k-th eigenvalue is given by λ_k = min_{dim(V)=k} max_{x ∈ V, x ≠ 0} R_u(x) = max_{dim(W)=n-k+1} min_{x ∈ W, x ≠ 0} R_u(x), where R_u(x) = ⟨u(x), x⟩ / ‖x‖^2 is the Rayleigh quotient.
6Using Burnside's Lemma, determine the number of distinct colorings of the 6 vertices of a regular planar hexagon using at most 2 colors (e.g. black and white), up to rotations and reflections belonging to the dihedral group D_6 of order 12.
A.9
B.13
C.16
D.24
Explanation: Burnside's lemma states |X/G| = (1/|G|) ∑_{g ∈ G} |X^g|. Here |G| = 12. The elements of D_6 act on vertices as follows: 1 identity element fixes all 2^6 = 64 colorings; 2 rotations by ±π/3 (cycle structure 6) fix 2^1 = 2 colorings each; 2 rotations by ±2π/3 (two 3-cycles) fix 2^2 = 4 colorings each; 1 rotation by π (three 2-cycles) fixes 2^3 = 8 colorings; 3 reflections across axes passing through opposite vertices (two fixed points, two 2-cycles, cycle structure 1^2 2^2) fix 2^4 = 16 colorings each; 3 reflections across axes passing through midpoints of opposite edges (three 2-cycles, cycle structure 2^3) fix 2^3 = 8 colorings each. Summing fixed points: 64 + 2(2) + 2(4) + 8 + 3(16) + 3(8) = 64 + 4 + 8 + 8 + 48 + 24 = 156. Dividing by |G| = 12 gives 156 / 12 = 13 orbits.
7In the Euclidean space R^3 equipped with the standard dot product, consider the subspace F = Vect(u_1, u_2) spanned by u_1 = (1, 0, 1) and u_2 = (1, 1, 0). What is the distance d(x, F) from the point x = (1, 2, 3) to F, computed using the Gram matrix G(u_1, u_2)?
A.1 / √3
B.√2 / 3
C.4 / √3
D.2 / √3
Explanation: The squared distance from x to F is d(x, F)^2 = det G(u_1, u_2, x) / det G(u_1, u_2). Alternatively, the unit normal to F is n = (u_1 × u_2) / ‖u_1 × u_2‖. We compute u_1 × u_2 = (1·0 - 1·1, 1·1 - 1·0, 1·1 - 0·1) = (-1, 1, 1). Its norm is √((-1)^2 + 1^2 + 1^2) = √3. The distance is the absolute value of the projection of x onto n: d(x, F) = |x · n| / ‖n‖ = |1(-1) + 2(1) + 3(1)| / √3 = |-1 + 2 + 3| / √3 = 4 / √3. Checking Gram determinants: det G(u_1, u_2) = (u_1 · u_1)(u_2 · u_2) - (u_1 · u_2)^2 = (2)(2) - (1)^2 = 3. det G(u_1, u_2, x) = [det(u_1, u_2, x)]^2 = (-1·1 + 1·2 + 1·3)^2 = 4^2 = 16. Thus d^2 = 16/3, so d = 4 / √3.
8What is the degree [Q(√2 + √3) : Q] of the field extension Q(√2 + √3) over Q, and what is the minimal polynomial of α = √2 + √3 over Q?
A.Degree 2, with minimal polynomial X^2 - 5
B.Degree 4, with minimal polynomial X^4 - 10X^2 + 25
C.Degree 2, with minimal polynomial X^2 - 2√6
D.Degree 4, with minimal polynomial X^4 - 10X^2 + 1
Explanation: Let α = √2 + √3. Squaring gives α^2 = 5 + 2√6, so α^2 - 5 = 2√6. Squaring again gives (α^2 - 5)^2 = 24, which expands to α^4 - 10α^2 + 25 = 24, so α^4 - 10α^2 + 1 = 0. The polynomial P(X) = X^4 - 10X^2 + 1 has roots ±√2 ± √3, none of which are rational. Furthermore, P(X) does not factor into rational quadratics (its quadratic factors over R are (X^2 - 2√2 X + 1) and (X^2 + 2√2 X + 1), which are not in Q[X]). Thus P(X) is irreducible over Q, and [Q(√2 + √3) : Q] = deg(P) = 4.
9Let S_4 be the symmetric group on 4 elements. What are the successive derived subgroups (derived series) D^1(S_4) = [S_4, S_4] and D^2(S_4) = [D^1(S_4), D^1(S_4)], and what does this establish regarding the solvability of S_4?
A.D^1(S_4) = A_4 and D^2(S_4) = V_4 (Klein 4-group); since D^3(S_4) = {id}, S_4 is solvable.
B.D^1(S_4) = A_4 and D^2(S_4) = A_4; since the series stabilizes at a non-trivial subgroup, S_4 is not solvable.
C.D^1(S_4) = V_4 and D^2(S_4) = {id}; S_4 is nilpotent of class 2.
D.D^1(S_4) = Z/2Z and D^2(S_4) = {id}; S_4 is a p-group.
Explanation: The commutator subgroup of S_n for n ≥ 2 is the alternating group A_n, so D^1(S_4) = A_4. The derived subgroup of A_4 is the Klein four-group V_4 = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, which is abelian. Because V_4 is abelian, its commutator subgroup D^3(S_4) = [V_4, V_4] = {id}. Because the derived series terminates at the trivial subgroup in finite steps, S_4 is a solvable group.
10Let J ∈ M_2(R) be the Jordan block J = [[2, 1], [0, 2]]. For any t ∈ R, what is the matrix exponential exp(t J)?
A.[[e^(2t), e^t], [0, e^(2t)]]
B.e^(2t) [[1, t], [0, 1]]
C.[[e^(2t), t e^t], [0, e^(2t)]]
D.e^(2t) [[1, 0], [t, 1]]
Explanation: We decompose J into J = 2 I_2 + N, where N = [[0, 1], [0, 0]]. Since 2 I_2 commutes with N, exp(t J) = exp(2t I_2) · exp(t N). We have exp(2t I_2) = e^(2t) I_2. Because N^2 = 0, the power series for exp(t N) terminates: exp(t N) = I_2 + t N = [[1, t], [0, 1]]. Multiplying by e^(2t) yields exp(t J) = e^(2t) [[1, t], [0, 1]] = [[e^(2t), t e^(2t)], [0, e^(2t)]].

About the Concours X-ENS-ESPCI Exam

The Concours X-ENS-ESPCI is France's premier and most intellectually demanding scientific competitive entrance examination. Governed by the decree of 13 October 2022 (arrêté du 13 octobre 2022), the common written bank (banque d'épreuves écrites communes) pools written examinations across six elite institutions: École Polytechnique ('l'X'), the four Écoles Normales Supérieures (ENS Paris-Ulm, ENS de Lyon, ENS Paris-Saclay, and ENS Rennes), and ESPCI Paris - PSL (in the PC stream). Candidates sit common written papers over five days across 33 examination centres (27 in metropolitan France, 3 in French overseas departments, and 3 international centres). While the written session is mutualised, each participating Grande École maintains full institutional autonomy: each school sets its own eligibility coefficients and admissibility thresholds (barres d'admissibilité), and conducts its own independent oral examinations (épreuves orales) in Palaiseau or on the respective ENS campuses. For the 2026 session, written examinations take place 13–17 April 2026 for MP, MPI, PC, and PSI (with PT candidates tested via Banque PT from 27 April to 7 May 2026); admissibility results are published on 27 May 2026; oral examinations run from 8 June to 12 July 2026; and final admission results appear on 24 July 2026 at 14:00. This practice bank provides 100 rigorous English-language multiple-choice questions adapted from the shared core CPGE curriculum, enabling candidates and international scholars to benchmark theoretical understanding, analytical derivation, and calculation speed.

Exam sponsor: Banque commune d'épreuves X-ENS-ESPCI (École Polytechnique, ENS Ulm, ENS de Lyon, ENS Paris-Saclay, ENS Rennes, ESPCI Paris). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written phase (écrits) of the common X-ENS-ESPCI bank, 13–17 April 2026, for the MP, MPI, PC and PSI streams, with one four-hour paper at 08:00 and one at 14:00 each day; PT candidates instead sit the Banque PT from 27 April to 7 May 2026. Papers typically comprise Mathématiques A and B, Physique, Français, Langue vivante and filière-specific papers (Informatique for MPI, Chimie for PC, Physique/Sciences de l'ingénieur for the SI option); some papers are used only by the ENS. Each institution applies its own coefficients and admissibility threshold: Polytechnique publishes admissibilité on 27 May 2026 and runs its own orals and sports tests at Palaiseau between 8 June and 12 July 2026 depending on filière, with admission results on 24 July 2026 from 14:00.

Time Limit

4 hours per paper (multi-day examination session)

Passing Score

Competitive ranking (classement) determined independently by each school's jury for admissibility (épreuves écrites) and final admission (épreuves orales).

Exam / Certification Fees

Polytechnique: 220 €; ESPCI Paris: 220 € (filière PC); ENS (Ulm, Lyon, Paris-Saclay, Rennes): free (0 €); boursiers de l'État: 0 €.

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

35%

Mathématiques Fondamentales et Analyse

Advanced linear algebra and duality, abstract group and ring theory, metric and normed space topology, complex analysis (Cauchy theorems, residues), Hilbert spaces and self-adjoint operators, differential geometry, and measure-theoretic probability.

25%

Physique Théorique et Appliquée

Covariant electrodynamics and potential formulations, statistical mechanics (microcanonical, canonical, grand canonical, quantum distributions), quantum observables and ladder operators, continuum mechanics and dispersive wave dynamics, and nonlinear stability.

15%

Chimie Fondamentale

Molecular orbital and Hückel theory, chemical thermodynamics (chemical potentials, activity coefficients, fugacity), organic reaction mechanisms (pericyclic reactions, organometallics), and interfacial electrochemical kinetics.

15%

Informatique Théorique et Algorithmique

Turing computability, decidability, computational complexity classes (P, NP, NP-complete reductions), formal grammars and automata, lambda calculus, and advanced algorithm design with amortized and flow analysis.

10%

Philosophie, Lettres et Langue Vivante

Epistemology and philosophy of science (falsifiability, paradigm shifts, underdetermination), formal argumentative structure, and critical English textual analysis.

Preparing for the Concours X-ENS-ESPCI Exam

What You Need to Know

  • Passing score: Competitive ranking (classement) determined independently by each school's jury for admissibility (épreuves écrites) and final admission (épreuves orales).
  • Assessment: Written phase (écrits) of the common X-ENS-ESPCI bank, 13–17 April 2026, for the MP, MPI, PC and PSI streams, with one four-hour paper at 08:00 and one at 14:00 each day; PT candidates instead sit the Banque PT from 27 April to 7 May 2026. Papers typically comprise Mathématiques A and B, Physique, Français, Langue vivante and filière-specific papers (Informatique for MPI, Chimie for PC, Physique/Sciences de l'ingénieur for the SI option); some papers are used only by the ENS. Each institution applies its own coefficients and admissibility threshold: Polytechnique publishes admissibilité on 27 May 2026 and runs its own orals and sports tests at Palaiseau between 8 June and 12 July 2026 depending on filière, with admission results on 24 July 2026 from 14:00.
  • Time limit: 4 hours per paper (multi-day examination session)
  • Exam / certification fees: Polytechnique: 220 €; ESPCI Paris: 220 € (filière PC); ENS (Ulm, Lyon, Paris-Saclay, Rennes): free (0 €); boursiers de l'État: 0 €. Official sources

Using Our Practice Resources

  • Work through all 100 available questions
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Concours X-ENS-ESPCI: Suggested Study Strategy

1Master foundational proofs from first principles: Examiners at X and ENS value theoretical rigor, precision in quantifier handling (∀, ∃), and complete mathematical hygiene over heuristic shortcuts.
2Work extensively through official annales: Practice solving 4-hour written problems from recent sessions of Mathématiques A, B, C and Physique A, B to internalize typical multi-part problem architecture.
3Cultivate speed in quantitative algebraic manipulation: Ensure you can diagonalize matrices, compute matrix exponentials, evaluate Cauchy contour integrals, and solve coupled differential equations accurately without relying on calculators.
4In physics, always conduct dimensional analysis (analyse dimensionnelle) and verify asymptotic behavior at extreme limits (e.g., t → 0, t → ∞, or k → 0) before accepting any derived formula.
5Review molecular orbital theory and thermodynamics: For PC candidates and cross-disciplinary papers, mastery of secular determinants, chemical potentials, and interfacial electrochemistry is decisive.
6Engage with the official français-philosophie yearly theme: A high mark on the 4-hour dissertation can significantly improve overall ranking, as language and culture carry non-negligible coefficients.

Frequently Asked Questions

What is the Concours X-ENS-ESPCI?

The Concours X-ENS-ESPCI is the joint written examination bank (banque commune d'épreuves d'admissibilité) through which École Polytechnique, the four Écoles Normales Supérieures (ENS Ulm, Lyon, Paris-Saclay, Rennes), and ESPCI Paris select candidates from French scientific preparatory classes (CPGE MP, MPI, PC, PSI). It represents the highest echelon of French scientific competitive examinations.

Which preparatory streams (filières) participate in the common written bank?

The written bank operates for candidates in MP (Mathématiques et Physique), MPI (Mathématiques, Physique et Informatique), PC (Physique et Chimie), and PSI (Physique et Sciences de l'Ingénieur). Candidates in the PT (Physique et Technologie) filière sit the separate Banque PT written session, with scores transmitted to École Polytechnique and the ENS.

How are admissibility and final admission determined across the partner institutions?

While the written papers are sat jointly under the common bank, there is no single merged jury. Each school applies its own specific coefficient weighting to the written papers to establish its own admissibility list (liste des admissibles). Candidates admitted to the oral stage then sit independent oral exams (épreuves orales) organized separately by École Polytechnique at Palaiseau and by the respective ENS.

What are the registration fees for the Concours X-ENS-ESPCI?

Registration is conducted centrally through SCEI (Service de Concours des Écoles d'Ingénieurs). For 2025/2026, the application fee is 220 € for École Polytechnique and 220 € for ESPCI Paris (PC filière). Application to all four ENS is free of charge (0 €). Candidates holding French government need-based scholarships (boursiers de l'État) are fully exempt from all registration fees (0 €).

What are the key dates for the 2026 session?

Registration takes place from 8 December 2025 to 12 January 2026 on scei-concours.fr. The common written examinations (épreuves écrites) are held from 13 to 17 April 2026. Admissibility results are announced on 27 May 2026. Oral examinations take place from 8 June to 12 July 2026, and final admission results are published on 24 July 2026 at 14:00.

What is the format of the official written examinations?

The official concours consists of multi-hour written problems—typically 4 hours per paper—requiring rigorous mathematical proofs, theoretical derivations, physical modeling, and essay writing in French. The competition does not use multiple-choice questions. This platform's 100-question question bank is an English-language MCQ study adaptation created for conceptual review and rapid quantitative testing.

Are programmable calculators allowed during the written tests?

Calculators are strictly prohibited in the mathematics, theoretical computer science, and French-philosophy papers. For physics and chemistry papers, calculators may be authorized if explicitly indicated on the paper cover, subject to the strict activation of 'mode examen' (cleared memory, flashing indicator).