Entrance examination X 25: mathematics B.
In this post you'll find suggested answers for the problem set B of the École Polytechnique competitive entrance exam 2025.
The aim is to study the existence and uniqueness, as well as some properties, of an osculating polynomial of minimal degree.
Given a function \(f\in\mathcal{C}^{\infty}([a,b])\), we look for a polynomial that coincides with the function and some of its successive derivatives at the points \(t_1,t_2,\dots,t_m\).
In the second part, we move to \(\mathbb{R}_n[X]\) fitted with the inner product \[ \begin{alignat*}{1} \langle P,Q\rangle &= \int_{-1}^1 P(x)Q(x)f(x)\mathrm{d}x \end{alignat*} \] where \[\begin{alignat*}{1} f:[-1,1]&\to\mathbb{R}^{+*}\\ \end{alignat*} \] a continuous function.
Here we construct a sequence of polynomials whose coordinates in the orthogonal basis obtained by applying Schmidt's procedure to the canonical basis \(1,X,X^2,\dots,X^n\) are positive numbers.
In the third part, we find the power series form of \[ \begin{alignat*}{1} r&\mapsto F_{\lambda}(x,r)=(1-2rx+r^2)^{-\lambda}\ \end{alignat*} \]
We obtain \[ \begin{alignat*}{1} F_{\lambda}(x,r)&=\sum_{n=0}^{+\infty} a_n(x)r^n\ \end{alignat*} \] where the $a_n$ are polynomials that form an orthogonal basis of \(\mathbb{R}[X]\) fitted with the scalar product: \[ \begin{alignat*}{1} \langle P,Q\rangle &= \int_{-1}^1 P(x)Q(x)(1-x^2)^{\lambda-\frac{1}{2}}\mathrm{d}x \end{alignat*} \]
In the final section, perhaps paradoxically the most accessible, we introduce the notion of a positive-type function in dimension \(N\): \(\forall k\in \mathbb{N}^*\), for any choice of \(k\) unit vectors of \(\mathbb{R}^N\), the symmetric matrix \(M\in\mathcal{M}_k(\mathbb{R})\) defined by \[ \begin{alignat*}{1} \forall (i,j) \in \lBrack 1,k \rBrack^{2},\quad m_{ij} &= f(x_i\cdot x_j) \end{alignat*} \] is positive semidefinite.
We use some of the results obtained in the first three parts.
The problem seems to me to be very difficult, with questions that border on the unfeasible right from the start of the problem.
Correction of the MP B problem set on April 15 for the X 2025 competition:
The problem statement: