An analysis-algebra problem that starts very quietly with the first 8 questions, then goes on with sequences of functions with values in \(R_n[X]\) whose coefficients are sums of integer series. Lots of algebra in the final section. The main difficulty of …

A difficult, demanding and very long algebra problem, which studies the quaternion field : matrix representation, descriptions of direct orthogonal endomorphisms, automorphisms. Many topics that are less frequently covered in other competitive exams: 1st y…

An analysis-algebra problem that studies the stability conditions of the following differential equation \[ \left\{ \begin{array}{ll} y'=\varphi(y)& \\ y(0)=x_0& \end{array} \right. \] where \(\varphi\) is a \(C^1\) map from \(\mathbb{R}^n\) to \(…

A rather interesting analysis problem that studies the Wallis function: \[ \begin{alignat*}{1} f:]-1,+\infty[&\to\mathbb{R}^{+*}\\ x&\mapsto \int_0^{\frac{\pi}{2}}(\sin t)^x \mathrm{d} t \end{alignat*} \] which is a generalization of the well-known integ…

A very long algebra problem set on polynomials and umbral calculus. At the heart of this formalism is the Pincherle derivation. We introduce the operator \(\mathbf{x}\) which is an endomorphism of polynomials: \[ \begin{alignat*}{1} \mathbf{x}:\math…