Centrale-Supélec 2026 — Mathématiques 1 (PSI)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A · I — Le cas n = 1 (Q1 à Q3)

   • Q1 — the distribution of X_1.
   • Q2 — its generating function, its expectation and its variance, given directly.
   • Q3 — the proof of these results.
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2. Exercise 2 — Partie A · II — Le cas n = 2 (Q4 à Q13)

   • Q4, Q5 — the first values, and the possible outcomes of the first two tosses when X_2 = k with k > 2.
   • Q6 — the second-order recurrence relation.
   • Q7 — the roots r_1 and r_2 of the associated quadratic.
   • Q8, Q9 — radius of convergence, then expression of P(X_2 = k).
   • Q10 — the probability of never obtaining two consecutive heads is zero.
   • Q11 to Q13 — generating function, existence of moments, expectation and variance.
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3. Exercise 3 — Partie B · I — Le cas n = 3 par l'algèbre linéaire (Q14 à Q25)

   • Q14, Q15 — first values and third-order recurrence.
   • Q16, Q17 — the companion matrix M and its characteristic polynomial.
   • Q18, Q19 — χ_M has a unique root in ]0,1[, and only one real root.
   • Q20 to Q23 — diagonalisation over C, general form of P(X_3 = k), and its real form.
   • Q24, Q25 — comparison of the moduli of the roots, then the asymptotic equivalent.
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4. Exercise 4 — Partie B · II — Un cas particulier : p = 3/4 (Q26 à Q28)

   • Q26 — an integer root of χ_M(X/4).
   • Q27 — all the roots of χ_M.
   • Q28 — an explicit asymptotic equivalent of P(X_3 = k).
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5. Exercise 5 — Partie C — Le cas général par la fonction génératrice (Q29 à Q35)

   f(t) = t^n − q·Σ_{i=1}^{n} p^{i−1}t^{n−i} is the function introduced in the paper.

   • Q29, Q30 — first values and recurrence of order n.
   • Q31 — f(1), and the existence of a t ∈ ]p, 1] such that f(t) ⩾ 0.
   • Q32 — the bound P(X_n = k) ⩽ t^k.
   • Q33, Q34 — the generating function of X_n, and its expectation.
   • Q35 — the partial fraction decomposition and how to use it.
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6. Exercise 6 — Partie D · I — Au moins n piles en k lancers (Q36 à Q40)

   A_{n,k} is the event "having obtained a run of at least n heads during the first k tosses".

   • Q36, Q37 — first values and recurrence.
   • Q38 — a Python function returning the list of the P(A_{n,l}).
   • Q39, Q40 — the constant solution, then switching to the sequence u_k = 1 − P(A_{n,k}).
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7. Exercise 7 — Partie D · II — La majoration géométrique et ses conséquences (Q41 à Q45)

   • Q41 — bounds on and monotonicity of (u_k).
   • Q42 — the bound u_k ⩽ α^{k−n} for some α ∈ [M, 1[.
   • Q43 — the probability of eventually obtaining n consecutive heads.
   • Q44 — a numerical application: five heads in a hundred tosses.
   • Q45 — the method to follow to obtain the closed form, without the computations.
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