PS 1.1★★★
Indicator functions, direct images and inverse images
Translate set operations into computations on functions, and characterise injectivity and surjectivity through direct and inverse images.
Problem sets — MPSI
Three problem sets per chapter, of increasing difficulty: ★★★ mastering the course, ★★★ standard competitive-exam level, ★★★ the most selective parts. Each problem set takes about 6 hours.
The documents (PDF) are written in French. Titles, objectives and quizzes are translated into English.
Go to a chapter
1. Logic and sets2. Complex numbers3. Algebraic manipulation4. Real numbers5. Sequences6. Functions of one real variable7. Antiderivatives and differential equations8. Arithmetic of integers9. Algebraic structures10. Matrices and linear systems11. Polynomials and rational functions12. Counting13. Probability14. Taylor expansions and asymptotics15. Linear algebra16. Symmetric group and determinants17. Inner product spaces18. Integration19. Improper integrals20. Series21. Functions of two variables
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1Logic and sets
PS 1.2★★★
Fixed points, closures and Galois connections
Prove the Knaster–Tarski fixed-point theorem and apply it to closure operators, Galois connections and inductive definitions.
PS 1.3★★★
Chains and antichains: the theorems of Mirsky and Dilworth
Prove the two dual decomposition theorems for a finite ordered set, then apply them to divisibility and to sequences of numbers.
2Complex numbers
PS 2.1★★★
Trigonometric sums, roots of unity and the value of $\sum\frac1{k^2}$
Master the computation of trigonometric sums, then compute $\sum_{k=1}^{n-1}\cotan^2\frac{k\pi}n$ and deduce, by squeezing, the value of $\lim_n\sum_{k=1}^n\frac1{k^2}$.
Problems PDF · 122 KBSolution on 20 November 2026
PS 2.2★★★
Triangle geometry: the nine-point circle and the Simson line
Study a triangle inscribed in the unit circle with complex numbers: orthocentre, Euler line, nine-point circle, Simson line.
Problems PDF · 110 KBSolution coming soon
PS 2.3★★★
Möbius transformations, cross-ratio and automorphisms of the unit disc
Study Möbius transformations of the complex plane, their action on circles and lines, the cross-ratio, then the transformations preserving the unit disc and a “distance” they preserve.
Problems PDF · 131 KBSolution coming soon
3Algebraic manipulation
PS 3.1★★★
Double sums and harmonic numbers
Handle simple and double sums with ease, through the study of the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$.
Problems PDF · 112 KBSolution coming soon
PS 3.2★★★
The Fibonacci sequence: identities and binomial coefficients
Prove the main Fibonacci identities by three complementary methods: induction, Binet's formula and binomial coefficients.
Problems PDF · 118 KBSolution coming soon
PS 3.3★★★
Classical inequalities and Carleman's inequality
Prove the great discrete inequalities (means, Cauchy–Schwarz, rearrangement, Chebyshev) by purely algebraic methods, then Carleman's inequality with its optimal constant $\e$.
Problems PDF · 119 KBSolution coming soon
4Real numbers
PS 4.1★★★
Integer part: identities, sums and counting
Learn the techniques of computation with the floor function and apply them to sums and to counting lattice points.
Problems PDF · 112 KBSolution coming soon
PS 4.2★★★
Rational approximation: Dirichlet's lemma and Liouville numbers
Measure “how well” a real number can be approximated by rationals, deduce an irrationality criterion and construct a very well approximable real number.
Problems PDF · 123 KBSolution coming soon
PS 4.3★★★
Continued fractions
Expand a real number as a continued fraction, establish the properties of the convergents and their optimality, and apply them to $\sqrt2$, the golden ratio, $\pi$ and the Pell–Fermat equation.
Problems PDF · 122 KBSolution coming soon
5Sequences
PS 5.1★★★
Recursive sequences: four complete studies
Carry out the full study of a sequence defined by $u_{n+1}=f(u_n)$: stable interval, monotonicity, limit, rate of convergence.
Problems PDF · 107 KBSolution coming soon
PS 5.2★★★
Computing fast: Heron, Newton and Archimedes' algorithm for $\pi$
Compare algorithms computing $\sqrt a$, $\frac1a$ and $\pi$ by their rate of convergence (linear, quadratic, cubic), and prove these rates.
Problems PDF · 110 KBSolution coming soon
PS 5.3★★★
The Toeplitz and Stolz–Cesàro theorems; speed of slowly converging recursive sequences
Establish a general averaging theorem, deduce the Stolz–Cesàro theorem, and use it to obtain the precise behaviour of slowly converging recursive sequences.
Problems PDF · 117 KBSolution coming soon
6Functions of one real variable
PS 6.1★★★
Intermediate values, Rolle's theorem and the Darboux property
Use the great theorems of real analysis (intermediate values, extreme values, Rolle) in varied situations, up to Darboux's theorem.
Problems PDF · 112 KBSolution coming soon
PS 6.2★★★
Functional equations: Jensen, d'Alembert, multiplicative Cauchy
Find all continuous functions satisfying a functional equation, by the method “values on a dense subset, then continuity”.
Problems PDF · 117 KBSolution coming soon
PS 6.3★★★
Convexity: Young, Hölder, Minkowski, log-convexity and Karamata's inequality
Derive the great inequalities of analysis from convexity, study log-convex functions, and prove Karamata's majorization inequality.
Problems PDF · 126 KBSolution coming soon
7Antiderivatives and differential equations
PS 7.1★★★
First-order equations: matching solutions, Bernoulli and Riccati
Solve first-order linear equations, including at points where the coefficient of $y'$ vanishes, then reduce classical non-linear equations to linear ones.
Problems PDF · 117 KBSolution coming soon
PS 7.2★★★
Oscillators: free motion, resonance and beats
Fully study the damped and driven harmonic oscillator, a common model in mechanics and electricity, and quantify the resonance phenomenon.
Problems PDF · 113 KBSolution coming soon
PS 7.3★★★
Sturm theory: zeros of the solutions of $y''+q(x)y=0$
Study qualitatively, without computing them, the solutions of a second-order linear equation with variable coefficient: separation of zeros, comparison, estimates of the gaps between zeros.
Problems PDF · 122 KBSolution coming soon
8Arithmetic of integers
PS 8.1★★★
Diophantine equations and infinite descent
Solve classical equations in integers (linear, quadratic, Fermat's equation for exponent $4$) by combining factorisation, congruences and infinite descent.
Problems PDF · 109 KBSolution coming soon
PS 8.2★★★
Arithmetic functions, Möbius inversion and perfect numbers
Study the multiplicative arithmetic functions ($\varphi$, $\mu$, $d$, $\sigma$) and Dirichlet convolution, and characterise the even perfect numbers.
Problems PDF · 116 KBSolution coming soon
PS 8.3★★★
Legendre's formula, Chebyshev's estimates and Bertrand's postulate
Study the distribution of prime numbers by elementary methods: bound $\pi(x)$ between two multiples of $\frac x{\ln x}$, and prove that there is always a prime between $n$ and $2n$.
Problems PDF · 123 KBSolution coming soon
9Algebraic structures
PS 9.1★★★
Transformation groups: the affine group, similarities and the isometries of the triangle
Use the vocabulary of groups on concrete transformation groups, and recognise isomorphisms.
Problems PDF · 108 KBSolution coming soon
PS 9.2★★★
Rings: Boolean rings, functions and $2\times2$ matrices
Explore, in three families of rings, the remarkable elements (units, idempotents, nilpotents, zero divisors), and construct $\C$ and $\Q(\sqrt2)$ as rings of matrices.
Problems PDF · 108 KBSolution coming soon
PS 9.3★★★
Quadratic fields and field homomorphisms
Study the subfields $\Q(\sqrt d)$, their isomorphisms and automorphisms, then prove that the identity is the only ring homomorphism from $\R$ to $\R$.
Problems PDF · 114 KBSolution coming soon
10Matrices and linear systems
PS 10.1★★★
Powers of matrices and coupled recursive sequences
Compute the powers of a matrix by several methods (binomial formula, annihilating polynomial, change of basis) and apply them to coupled sequences.
Problems PDF · 111 KBSolution coming soon
PS 10.2★★★
Jacobi and Gauss–Seidel iterative methods
Study two iterative methods for solving $Ax=b$, establish a convergence criterion (strictly diagonally dominant matrices) and estimate the rate of convergence.
Problems PDF · 117 KBSolution coming soon
PS 10.3★★★
Rank-one matrices, Sherman–Morrison and Woodbury formulas
Study rank-one matrices and “low-rank updates” of an invertible matrix, a central tool of scientific computing (recursive least squares, changing one coefficient).
Problems PDF · 113 KBSolution coming soon
11Polynomials and rational functions
PS 11.1★★★
Chebyshev polynomials
Construct the Chebyshev polynomials from the formula $T_n(\cos\theta)=\cos(n\theta)$ and establish their algebraic and analytic properties.
Problems PDF · 116 KBSolution coming soon
PS 11.2★★★
Bernoulli polynomials and sums of powers
Construct the Bernoulli polynomials, establish their symmetry properties, and deduce formulas for the sums $\sum_{k=0}^{m-1}k^p$.
Problems PDF · 101 KBSolution coming soon
PS 11.3★★★
Cyclotomic polynomials
Study the cyclotomic polynomials $\Phi_n$, show that they have integer coefficients, prove that $\Phi_p$ is irreducible, and prove that there are infinitely many primes congruent to $1$ modulo $n$.
Problems PDF · 123 KBSolution coming soon
12Counting
PS 12.1★★★
Lattice paths and binomial identities
Prove binomial identities by combinatorial arguments (double counting, bijections) using lattice paths.
Problems PDF · 128 KBSolution coming soon
PS 12.2★★★
Stirling numbers, surjections and Bell numbers
Count the partitions of a finite set and the surjections, and relate these numbers to polynomials and to the exponential series.
Problems PDF · 115 KBSolution coming soon
PS 12.3★★★
Integer partitions: Euler, Glaisher and Franklin
Study integer partitions through bijections: conjugation of Ferrers diagrams, Euler's theorem (odd parts and distinct parts), and Euler's pentagonal number theorem via Franklin's involution.
Problems PDF · 124 KBSolution coming soon
13Probability
PS 13.1★★★
Conditioning and the matching problem
Master Bayes' formula, then study the distribution of the number of fixed points of a random permutation and its approximation by a Poisson distribution.
Problems PDF · 114 KBSolution coming soon
PS 13.2★★★
Simple random walk on $\Z$: reflection and returns to the origin
Study the simple random walk: probability of returning to $0$, reflection principle, distribution of the first return time, and recurrence or transience depending on $p$.
Problems PDF · 111 KBSolution coming soon
PS 13.3★★★
Concentration and the Weierstrass approximation theorem
Compare concentration inequalities for sums of Bernoulli variables (Bienaymé–Chebyshev, Hoeffding), and prove the Weierstrass theorem for Lipschitz functions using Bernstein polynomials.
Problems PDF · 125 KBSolution coming soon
14Taylor expansions and asymptotics
PS 14.1★★★
Taylor expansions: computations and local study
Learn the technique of Taylor expansions and apply it to limits, tangents and asymptotes.
Problems PDF · 102 KBSolution coming soon
PS 14.2★★★
Asymptotic expansions of sums
Obtain precise asymptotic expansions of classical sums ($H_n$, $\sum\frac1{k^2}$, $\sum\ln k$, $\sum\frac1{\sqrt k}$) by the “corrected telescoping” method.
Problems PDF · 113 KBSolution coming soon
PS 14.3★★★
Order of convergence of iterative methods and acceleration
Measure the rate of convergence of iterative methods (fixed point, Newton, secant) using Taylor expansions, then accelerate slow convergence (Aitken's $\Delta^2$ process, Richardson extrapolation).
Problems PDF · 119 KBSolution coming soon
15Linear algebra
PS 15.1★★★
Lagrange interpolation, linear forms and quadrature formulas
Use the isomorphism between polynomials and values at points to build approximate integration formulas that are exact on polynomials of given degree (rectangle, trapezoidal and Simpson's rules).
Problems PDF · 125 KBSolution coming soon
PS 15.2★★★
Nilpotent endomorphisms
Establish the properties of nilpotent endomorphisms: index, sequence of ranks, cyclic bases, reduction to a “Jordan” matrix, and commutant.
Problems PDF · 99 KBSolution coming soon
PS 15.3★★★
Differentiation and difference operators on $\K[X]$
Study differentiation $D$, translation $\tau$ and the difference operator $\Delta=\tau-\Id$ on polynomials, the Newton binomial basis, and the “exponential” and “logarithm” relations between them.
Problems PDF · 112 KBSolution coming soon
16Symmetric group and determinants
PS 16.1★★★
Vandermonde and circulant determinants, with applications
Compute the Vandermonde determinant and its variants, then apply it to interpolation, to circulant matrices and to the linear independence of families of functions.
Problems PDF · 117 KBSolution coming soon
PS 16.2★★★
The symmetric group: generators, inversions and sorting
Study various generating sets of $\Sg_n$, relate the number of inversions of a permutation to sorting by adjacent transpositions, and count permutations by number of inversions or cycles.
Problems PDF · 119 KBSolution coming soon
PS 16.3★★★
The resultant of two polynomials
Construct the resultant of two polynomials, characterise the existence of a common root by the vanishing of a determinant, and deduce the discriminant and elimination methods.
Problems PDF · 118 KBSolution coming soon
17Inner product spaces
PS 17.1★★★
The Frobenius inner product and the least-squares line
Work with the canonical inner product of $\Mat_n(\R)$ and derive matrix inequalities from it, then interpret linear regression as an orthogonal projection.
Problems PDF · 115 KBSolution coming soon
PS 17.2★★★
Orthogonal polynomials: three-term recurrence, zeros and Gaussian quadrature
Construct the orthogonal polynomials associated with a weight, establish their recurrence relation and the location of their zeros, and deduce the Gaussian quadrature formulas.
Problems PDF · 118 KBSolution coming soon
PS 17.3★★★
Trigonometric polynomials: Bessel, Fejér, Parseval and the isoperimetric inequality
Study the approximation of periodic functions by trigonometric polynomials, prove Parseval's identity for Lipschitz functions using the Fejér kernel, and deduce Wirtinger's inequality and the isoperimetric inequality.
Problems PDF · 123 KBSolution coming soon
18Integration
PS 18.1★★★
Riemann sums and elementary integral inequalities
Compute limits of sums and products using Riemann sums, and establish classical integral inequalities.
Problems PDF · 105 KBSolution coming soon
PS 18.2★★★
Convexity and integrals: Hermite–Hadamard, Jensen, Hölder
Pass from discrete convexity inequalities to their integral versions, and apply them to bounds on integrals and means.
Problems PDF · 115 KBSolution coming soon
PS 18.3★★★
Irrationality of $\pi^2$ and $\e^r$ (Niven's method)
Prove, by elementary integration methods, that $\pi^2$ is irrational (hence so is $\pi$), and that $\e^r$ is irrational for every non-zero rational $r$.
Problems PDF · 113 KBSolution coming soon
19Improper integrals
PS 19.1★★★
Convergence of integrals: Bertrand, $\frac{\sin t}{t^\alpha}$ and Fresnel
Determine the convergence of classical integrals, including conditionally convergent ones, and master the comparison tools.
Problems PDF · 115 KBSolution coming soon
PS 19.2★★★
Frullani integrals and classical computations
Establish Frullani's theorem and compute classical integrals ($\int_0^{\pi/2}\ln\sin$, $\int_0^\infty\frac{\ln(1+t^2)}{1+t^2}$, …) by elementary manipulations.
Problems PDF · 114 KBSolution coming soon
PS 19.3★★★
The Laplace transform: Abelian theorems and injectivity
Study the Laplace transform of a continuous function, its behaviour at $0$ and at $+\infty$ (Abelian theorems), and prove that it characterises the function.
Problems PDF · 117 KBSolution coming soon
20Series
PS 20.1★★★
Alternating series and oscillating terms
Master the alternating series test and its bound on the remainder, study “almost alternating” series by asymptotic expansion, and handle classical oscillating terms from competitive exams.
Problems PDF · 122 KBSolution coming soon
PS 20.2★★★
Summation of series of rational functions
Characterise the convergence of series of rational functions, introduce the function $H(x)=\sum_{n\ge1}\bigl(\frac1n-\frac1{n+x}\bigr)$ that extends the sequence $(H_n)$, and use it to compute sums explicitly.
Problems PDF · 123 KBSolution coming soon
PS 20.3★★★
Abel summation and the trigonometric series $\sum\frac{\sin(nx)}n$
Prove the Abel–Dirichlet test, study the series $\sum\frac{\e^{\i n\theta}}{n^\alpha}$, then compute the sums $\sum\frac{\sin(nx)}n$, $\sum\frac{\cos(nx)}n$, $\sum\frac{1-\cos(nx)}{n^2}$ using the Riemann–Lebesgue lemma, with applications to $\zeta(2)$ and the identity $\sum\frac{\sin n}n=\sum\frac{\sin^2n}{n^2}=\frac{\pi-1}2$.
Problems PDF · 128 KBSolution coming soon
21Functions of two variables
PS 21.1★★★
First-order partial differential equations
Check that a piecewise-defined function is of class $\mathcal C^1$, then solve first-order partial differential equations by a suitable change of variables; recognise the vector fields that are gradients.
Problems PDF · 124 KBSolution coming soon
PS 21.2★★★
Roulettes: cycloid, epicycloids and hypocycloids
Study the curves traced by a point of a circle rolling without slipping on a line or a circle, analyse their cusps and establish some remarkable geometric properties (normal through the contact point, chords of constant length of the cardioid, tangents of the deltoid).
Problems PDF · 129 KBSolution coming soon
PS 21.3★★★
Convex functions of two variables and gradient descent
Characterise convex functions of class $\mathcal C^1$ on $\R^2$ by their gradient, then prove the geometric convergence of fixed-step gradient descent for strongly convex functions with Lipschitz gradient, and study the optimal step in the quadratic case.
Problems PDF · 152 KBSolution coming soon