MathDB

1

Part of 2023 Francophone Mathematical Olympiad

Problems(2)

4 b_{n+1} <= P(1)^2

Source: P1 Francophone Math Olympiad Senior 2023

5/2/2023
Let P(X)=anXn+an−1Xn−1+⋯+a1X+a0P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0 be a polynomial with real coefficients such that 0⩽ai⩽a00 \leqslant a_i \leqslant a_0 for i=1,2,…,ni = 1, 2, \ldots, n. Prove that, if P(X)2=b2n X2n +b2n−1X2n−1+⋯+bn+1Xn+1+⋯+b1X+b0P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0, then 4bn+1⩽P(1)24 b_{n+1} \leqslant P(1)^2.
polynomialalgebrainequalities
u_{2023} wanted, u_{k+2} >= 2 + u_K, u_{\ell+5} <=5 + u_\ell

Source: P1 Francophone Math Olympiad Junior 2023

5/2/2023
Let u0,u1,u2,…u_0, u_1, u_2, \ldots be integers such that u0=100u_0 = 100; uk+2⩾2+uku_{k+2} \geqslant 2 + u_k for all k⩾0k \geqslant 0; and uℓ+5⩽5+uℓu_{\ell+5} \leqslant 5 + u_\ell for all ℓ⩾0\ell \geqslant 0. Find all possible values for the integer u2023u_{2023}.
inequalitiesnumber theorySequence