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Key Facts: Concours Commun Mines-Ponts Exam

10 Schools

Prestigious engineering institutions recruited via a single common contest

Notice CCMP 2026

360 €

Unified registration fee (free for CROUS scholarship recipients)

Notice des Écoles CCMP 2026

14,364 Candidates

Registered applicants competing for 1,569 total places (2025 session)

Rapport Général CCMP 2025

4 Days

Duration of written examinations (27-30 April 2026)

Calendrier SCEI / CCMP 2026

The Concours Commun Mines-Ponts (CCMP) is the elite competitive examination recruiting for 10 top French engineering schools. Written papers (écrits) run over 4 days from 27 to 30 April 2026, followed by oraux from 22 June to 18 July 2026. This 100-question practice bank is an English-language MCQ study adaptation covering core mathematics, physics, chemistry, computer science/engineering, and humanities across the CPGE syllabus.

Sample Concours Commun Mines-Ponts Practice Questions

Try these sample questions to review concepts for the Concours Commun Mines-Ponts exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1In a real pre-Hilbert space $(E, \langle \cdot, \cdot \rangle)$ with induced norm \|\cdot\|, let $x, y \in E$ be non-zero vectors. Under what exact condition does the Cauchy-Schwarz equality |\langle x, y \rangle| = \|x\| \|y\| hold?
A.The vectors $x$ and $y$ are linearly dependent (collinear)
B.The vectors $x$ and $y$ are orthogonal, satisfying $\langle x, y \rangle = 0$
C.The vectors $x$ and $y$ have identical Euclidean norms, \|x\| = \|y\|
D.The sum vector satisfies \|x + y\|^2 = \|x\|^2 + \|y\|^2
Explanation: In any real pre-Hilbert space, the quadratic polynomial $P(t) = \|tx + y\|^2 = t^2 \|x\|^2 + 2t\langle x, y \rangle + \|y\|^2$ is non-negative for all $t \in \mathbb{R}$. Its discriminant $\Delta = 4(\langle x, y \rangle^2 - \|x\|^2 \|y\|^2)$ must be non-positive, yielding the Cauchy-Schwarz inequality. The equality |\langle x, y \rangle| = \|x\| \|y\| occurs if and only if $\Delta = 0$, which corresponds to the existence of a real root $t_0$ such that \|t_0 x + y\| = 0$, proving that $x$ and $y$ are linearly dependent.
2Let $A \in S_n^{++}(\mathbb{R})$ be a real symmetric positive-definite matrix and let $B \in S_n(\mathbb{R})$ be a real symmetric matrix. According to the simultaneous reduction theorem (théorème de réduction simultanée), which of the following statements is unconditionally true?
A.There exists an orthogonal matrix $Q \in O_n(\mathbb{R})$ such that both $Q^T A Q$ and $Q^T B Q$ are diagonal
B.There exists an invertible matrix $P \in GL_n(\mathbb{R})$ such that $P^T A P = I_n$ and $P^T B P$ is diagonal
C.The matrices $A$ and $B$ must commute ($AB = BA$) for simultaneous congruence to hold
D.There exists an invertible matrix $P \in GL_n(\mathbb{R})$ such that $P^{-1} A P$ and $P^{-1} B P$ are both the identity matrix
Explanation: Because $A$ is symmetric positive-definite, it defines an inner product on $\mathbb{R}^n$. By the spectral theorem applied to the self-adjoint endomorphism associated with the quadratic form of $B$ relative to this inner product, there exists a basis orthonormal for $A$ and orthogonal for $B$. In matrix terms, there exists $P \in GL_n(\mathbb{R})$ such that $P^T A P = I_n$ and $P^T B P = D$ where $D$ is real diagonal.
3In the pre-Hilbert space $E = C^0([-1, 1], \mathbb{R})$ equipped with the inner product $\langle f, g \rangle = \int_{-1}^1 f(t)g(t) \, dt$, what is the orthogonal projection of the polynomial function $h(t) = t^2$ onto the subspace $F = \mathbb{R}_1[X]$ of polynomial functions of degree at most 1?
A.$p_F(h)(t) = 0$
B.$p_F(h)(t) = t$
C.$p_F(h)(t) = \frac{1}{3}$
D.$p_F(h)(t) = \frac{1}{2} + t$
Explanation: The subspace $F$ has orthogonal basis $\{e_0, e_1\}$ where $e_0(t) = 1$ and $e_1(t) = t$ because $\langle 1, t \rangle = \int_{-1}^1 t \, dt = 0$. The orthogonal projection is $p_F(h) = \frac{\langle h, 1 \rangle}{\|1\|^2} 1 + \frac{\langle h, t \rangle}{\|t\|^2} t$. We compute $\langle h, 1 \rangle = \int_{-1}^1 t^2 \, dt = \frac{2}{3}$, $\|1\|^2 = \int_{-1}^1 1 \, dt = 2$, and $\langle h, t \rangle = \int_{-1}^1 t^3 \, dt = 0$ by parity. Thus, $p_F(h)(t) = \frac{2/3}{2} \cdot 1 + 0 \cdot t = \frac{1}{3}$.
4Let $H$ be a real or complex Hilbert space and let $u \in \mathcal{L}(H)$ be a continuous linear operator with adjoint $u^*$. Which relation correctly characterizes the orthogonal complement of the image of $u$?
A.$(\text{Im}\, u)^\perp = \text{Im}(u^*)$
B.$(\text{Im}\, u)^\perp = \ker(u)$
C.$(\text{Im}\, u)^\perp = (\ker(u^*))^\perp$
D.$(\text{Im}\, u)^\perp = \ker(u^*)$
Explanation: For any $y \in H$, $y \in (\text{Im}\, u)^\perp \iff \forall x \in H, \langle u(x), y \rangle = 0 \iff \forall x \in H, \langle x, u^*(y) \rangle = 0 \iff u^*(y) = 0 \iff y \in \ker(u^*)$. Hence, $(\text{Im}\, u)^\perp = \ker(u^*)$ holds in any Hilbert space.
5Consider the smooth function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x, y) = x^3 + y^3 - 3xy$. What is the nature of the critical point located at $(1, 1)$?
A.A strict local minimum with critical value $f(1, 1) = -1$
B.A saddle point (point col) because the Hessian has mixed signs
C.A strict local maximum with critical value $f(1, 1) = -1$
D.A degenerate non-isolated critical point with zero Hessian determinant
Explanation: The gradient is $\nabla f(x, y) = (3x^2 - 3y, 3y^2 - 3x)$, which vanishes at $(0, 0)$ and $(1, 1)$. The Hessian matrix of second partial derivatives is $H_f(x, y) = \begin{pmatrix} 6x & -3 \\ -3 & 6y \end{pmatrix}$. At $(1, 1)$, $H_f(1, 1) = \begin{pmatrix} 6 & -3 \\ -3 & 6 \end{pmatrix}$. Its determinant is $\det(H_f) = 36 - 9 = 27 > 0$, and the trace is $\text{Tr}(H_f) = 12 > 0$ with $f_{xx} = 6 > 0$. Thus, $H_f(1, 1)$ is symmetric positive-definite, proving $(1, 1)$ is a strict local minimum with $f(1, 1) = 1 + 1 - 3 = -1$.
6Using the method of Lagrange multipliers, what is the minimum value of $f(x, y, z) = x^2 + y^2 + z^2$ subject to the affine constraint $x + 2y + 3z = 14$ on $\mathbb{R}^3$?
A.7
B.14
C.28
D.196
Explanation: Let $g(x, y, z) = x + 2y + 3z - 14 = 0$. The Lagrange multiplier condition $\nabla f = \lambda \nabla g$ gives $(2x, 2y, 2z) = \lambda(1, 2, 3)$, so $x = \lambda/2$, $y = \lambda$, $z = 3\lambda/2$. Substituting into the constraint gives $\frac{\lambda}{2} + 2\lambda + \frac{9\lambda}{2} = 7\lambda = 14$, so $\lambda = 2$. This yields the minimizer $(x, y, z) = (1, 2, 3)$. The minimum value is $f(1, 2, 3) = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$. Geometrically, this is the square of the distance from the origin to the plane: $\frac{|-14|^2}{1^2 + 2^2 + 3^2} = \frac{196}{14} = 14$.
7Let $S$ be the two-dimensional submanifold of $\mathbb{R}^3$ defined by the level set $S = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 - z^2 = 1\}$. What is the Cartesian equation of the affine tangent plane to $S$ at the point $M_0 = (1, 1, 1)$?
A.$x + y + z = 3$
B.$x + y - z = 0$
C.$x + y - z = 1$
D.$2x + 2y - z = 3$
Explanation: The surface is defined as $g(x, y, z) = 1$ with $g(x, y, z) = x^2 + y^2 - z^2$. The gradient vector is $\nabla g(x, y, z) = (2x, 2y, -2z)$. At $M_0 = (1, 1, 1)$, $\nabla g(1, 1, 1) = (2, 2, -2)$, which is non-zero, confirming $(1, 1, 1)$ is a regular point. The affine tangent plane equation is $\nabla g(1, 1, 1) \cdot ((x, y, z) - (1, 1, 1)) = 0 \iff 2(x - 1) + 2(y - 1) - 2(z - 1) = 0 \iff 2x + 2y - 2z = 2 \iff x + y - z = 1$.
8Consider the sequence of functions $(f_n)_{n \ge 1}$ defined on $[0, 1]$ by $f_n(x) = \frac{nx}{1 + n^2 x^2}$. Which statement correctly describes the convergence of $(f_n)$ on $[0, 1]$ and on compact subintervals $[a, 1]$ with $a > 0$?
A.$(f_n)$ converges uniformly to 0 on $[0, 1]$
B.$(f_n)$ converges pointwise to 0 on $[0, 1]$, but does not converge uniformly on any interval $[a, 1]$ with $a > 0$
C.$(f_n)$ converges uniformly on $[0, 1]$ to the discontinuous function $f(x) = 1/2$ for $x > 0$ and $f(0) = 0$
D.$(f_n)$ converges pointwise to 0 on $[0, 1]$, does not converge uniformly on $[0, 1]$, but converges uniformly on $[a, 1]$ for every $a > 0$
Explanation: For $x = 0$, $f_n(0) = 0$. For $x > 0$, $f_n(x) \sim \frac{nx}{n^2 x^2} = \frac{1}{nx} \to 0$ as $n \to \infty$. So $(f_n)$ converges pointwise to the zero function $f \equiv 0$ on $[0, 1]$. However, \|f_n\|_\infty = \sup_{x \in [0, 1]} |f_n(x)| = f_n(1/n) = \frac{1}{1 + 1} = \frac{1}{2} \not\to 0$, so convergence is not uniform on $[0, 1]$. For any $a > 0$, once $n > 1/a$, the function $f_n$ is strictly decreasing on $[a, 1]$, so \sup_{x \in [a, 1]} |f_n(x)| = f_n(a) = \frac{na}{1 + n^2 a^2} \to 0$, ensuring uniform convergence on $[a, 1]$.
9What is the radius of convergence $R$ of the complex power series $\sum_{n=1}^\infty \frac{(2n)!}{(n!)^2} z^n$?
A.$R = \frac{1}{4}$
B.$R = \frac{1}{2}$
C.$R = 1$
D.$R = 4$
Explanation: Let $a_n = \frac{(2n)!}{(n!)^2} = \binom{2n}{n}$. We apply d'Alembert's ratio test: $\frac{a_{n+1}}{a_n} = \frac{(2n+2)!}{((n+1)!)^2} \frac{(n!)^2}{(2n)!} = \frac{(2n+2)(2n+1)}{(n+1)^2} = \frac{2(2n+1)}{n+1} \to 4$ as $n \to \infty$. Therefore, the radius of convergence is $R = \frac{1}{\lim_{n \to \infty} \frac{a_{n+1}}{a_n}} = \frac{1}{4}$.
10Consider the series of functions $\sum_{n=1}^\infty u_n(x)$ where $u_n(x) = \frac{\cos(nx)}{n^2 + x^2}$ defined on $\mathbb{R}$. Which of the following statements is strictly correct regarding its convergence and sum function $S(x)$?
A.The series converges pointwise on $\mathbb{R}$, but fails to converge uniformly because $n^2 + x^2$ depends on $x$
B.The series converges normally on $\mathbb{R}$, and its sum function $S$ is continuous and bounded on $\mathbb{R}$
C.The series converges normally on every compact interval, but not on $\mathbb{R}$
D.The series does not converge absolutely for $x = 0$
Explanation: For all $x \in \mathbb{R}$ and all $n \ge 1$, we have $|u_n(x)| = \frac{|\cos(nx)|}{n^2 + x^2} \le \frac{1}{n^2 + x^2} \le \frac{1}{n^2}$. Thus, the uniform norm satisfies \|u_n\|_\infty = \sup_{x \in \mathbb{R}} |u_n(x)| \le \frac{1}{n^2}$. Because the numerical series \sum_{n=1}^\infty \frac{1}{n^2} converges (Riemann series with $\alpha = 2 > 1$), the series \sum u_n converges normally (convergence normale) on all of $\mathbb{R}$. Normal convergence implies uniform convergence; since each $u_n$ is continuous and bounded, the sum $S(x)$ is continuous and bounded on $\mathbb{R}$.

About the Concours Commun Mines-Ponts Exam

The Concours Commun Mines-Ponts (CCMP) is one of France's premier competitive entrance examinations for students enrolled in Classes Préparatoires aux Grandes Écoles (CPGE) across the scientific tracks (MP, MPI, PC, and PSI). It grants admission to ten elite engineering institutions: École des Ponts ParisTech, Mines Paris-PSL, ISAE-SUPAERO, ENSTA Paris, Télécom Paris, IMT Atlantique, ENSAE Paris, Chimie ParisTech - PSL, Mines Saint-Étienne, and Mines Nancy. The examination comprises four intense days of written papers (écrits) held at major national examination centers (notably Paris Nord Villepinte, ~40 provincial cities, 7 DOM centers, and 9 international sites), followed by rigorous oral examinations (oraux) and the TIPE defense for admissible candidates. Note: Official CCMP examinations consist of extensive multi-page analytical problems written in French where calculators are forbidden. This 100-question practice bank is an English-language study adaptation engineered for conceptual mastery, active retrieval, and formula retention across the shared syllabus.

Exam sponsor: Direction du Concours Commun Mines-Ponts (GIP CCMP). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written phase (écrits) over 4 consecutive days (27-30 April 2026) consisting of 7-8 written papers (3-4 hours each) across Mathematics (I & II), Physics (I & II), Chemistry, Computer Science / Industrial Engineering, Français-Philosophie, and Modern Language. Admissible candidates proceed to oral examinations (oraux) from 22 June to 18 July 2026, including school-specific oral interrogations and the TIPE presentation.

Time Limit

Written écrits: 4 consecutive days (3 to 4 hours per paper, typically 7-8 written papers total). Oral examinations: 20-30 minutes per subject with 30-minute preparation.

Passing Score

Competitive selection based on national rank (rang de classement). Admissibilité threshold (barre d'admissibilité) is set per school and filière; any mark strictly below 3/20 in Français-Philosophie eliminates the candidate. Admission is determined by total weighted points (écrits + oraux).

Exam / Certification Fees

360 € single unified registration fee for all 10 member schools (completely free for boursiers du gouvernement français / CROUS scholarship holders).

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.

30%

Mathématiques

Pre-Hilbert and Hilbert spaces, spectral theorem for compact and self-adjoint operators, multivariable calculus (differentials, extrema, submanifolds), series of functions, Fourier transforms, discrete and continuous probability models, central limit theorem.

25%

Physique

Classical field theory, Maxwell's equations and relativistic invariants, electromagnetic radiation and antenna theory, fluid mechanics (Navier-Stokes, vorticity, boundary layers), wave propagation in anisotropic media, statistical physics (canonical ensemble), quantum mechanics (harmonic oscillator, spin).

15%

Chimie

Quantum chemistry and atomic orbitals, coordination chemistry and crystal field theory, chemical thermodynamics (chemical potentials, binary phase diagrams, Gibbs-Duhem), electrochemical kinetics (Butler-Volmer), modern organic reaction mechanisms.

15%

Informatique et Sciences Industrielles

Algorithm design and asymptotic complexity, graph algorithms, dynamic programming, formal languages and regular expressions, multi-domain system modeling, frequency response and state-space control.

15%

Culture Générale et Langue Vivante

Deep philosophical text analysis, critical argumentation, epistemological concepts, and high-level English comprehension and analytical reasoning.

Preparing for the Concours Commun Mines-Ponts Exam

What You Need to Know

  • Passing score: Competitive selection based on national rank (rang de classement). Admissibilité threshold (barre d'admissibilité) is set per school and filière; any mark strictly below 3/20 in Français-Philosophie eliminates the candidate. Admission is determined by total weighted points (écrits + oraux).
  • Assessment: Written phase (écrits) over 4 consecutive days (27-30 April 2026) consisting of 7-8 written papers (3-4 hours each) across Mathematics (I & II), Physics (I & II), Chemistry, Computer Science / Industrial Engineering, Français-Philosophie, and Modern Language. Admissible candidates proceed to oral examinations (oraux) from 22 June to 18 July 2026, including school-specific oral interrogations and the TIPE presentation.
  • Time limit: Written écrits: 4 consecutive days (3 to 4 hours per paper, typically 7-8 written papers total). Oral examinations: 20-30 minutes per subject with 30-minute preparation.
  • Exam / certification fees: 360 € single unified registration fee for all 10 member schools (completely free for boursiers du gouvernement français / CROUS scholarship holders). Official sources

Using Our Practice Resources

  • Work through all 100 available questions
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Concours Commun Mines-Ponts: Suggested Study Strategy

1Practice Without a Calculator: Because calculators are forbidden at CCMP, train mental arithmetic, logarithmic Taylor expansions (ln(1+x) ≈ x), and standard trigonometric values daily.
2Master the Core Spectral & Analytical Theorems: Pre-Hilbert geometry, symmetric compact operators, differential extrema under constraints, and Fourier transforms form the backbone of Mathematics I and II.
3Balance Mathematics with Français and Modern Language: Remember that a grade below 3/20 in Français-Philosophie eliminates you completely; dedicate regular weekly time to essay outlines and philosophical citation synthesis.
4Emphasize Dimensional Analysis & Boundary Layers: In physics and fluid dynamics, always verify physical dimensions and examine limiting cases (e.g., Reynolds number Re -> 0 or Re -> infinity) before finalizing calculations.
5Work with Timed Past Problems: Use official CCMP annales to build stamina for 4-hour sessions where problem length exceeds what any candidate can fully complete.

Frequently Asked Questions

What is the Concours Commun Mines-Ponts (CCMP)?

The CCMP is a unified French competitive examination (concours) allowing students from scientific preparatory classes (CPGE filières MP, MPI, PC, and PSI) to compete for entry into 10 prestigious engineering schools: École des Ponts ParisTech, Mines Paris-PSL, ISAE-SUPAERO, ENSTA Paris, Télécom Paris, IMT Atlantique, ENSAE Paris, Chimie ParisTech - PSL, Mines Saint-Étienne, and Mines Nancy.

What are the key dates and registration fees for the 2026 CCMP session?

Registration opens on 8 December 2025 and closes on 12 January 2026 on the SCEI platform. The unified registration fee is 360 € for non-scholarship candidates, and completely free (0 €) for CROUS/state scholarship holders (boursiers). Written exams take place from 27 to 30 April 2026. Admissibility results are released on 3 June 2026 at 18:00, orals are conducted from 22 June to 18 July 2026, admission lists appear on 22 July 2026 at 18:00, and the first SCEI admission proposals occur on 28 July 2026.

Are calculators allowed during the CCMP written examinations?

No. Under official CCMP regulations, calculators of any kind are strictly forbidden in all written papers (calculatrices interdites). Candidates must carry out all algebraic manipulations, numerical approximations, and order-of-magnitude estimates by hand.

Is there an eliminatory grade on the CCMP?

Yes. A mark strictly lower than 3 out of 20 (note < 3/20) in the Français-Philosophie written examination is strictly eliminatory for admissibilité, regardless of performance in mathematics or physics.

How does this question bank relate to the real CCMP examination?

The real CCMP examination consists of multi-hour written problem papers (épreuves écrites de 3 ou 4 heures) in French featuring complex multi-stage proofs, derivations, and essay questions. This 100-question practice bank is an English-language multiple-choice study adaptation designed to test core theoretical foundations, quantitative mathematical calculations, physical modeling, and algorithmic reasoning from the shared CPGE curriculum.