MathDB

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Part of 2022 Vietnam National Olympiad

Problems(2)

VMO 2022 problem 1 day 1

Source: Vietnam Mathematical Olympiad 2022 problem 1 day 1

3/4/2022
Let aa be a non-negative real number and a sequence (un)(u_n) defined as: u1=6,un+1=2n+an+n+anun+4,∀n≥1u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1 a) With a=0a=0, prove that there exist a finite limit of (un)(u_n) and find that limit b) With a≥0a \ge 0, prove that there exist a finite limit of (un)(u_n)
vmocalculusSequencealgebra
VMO 2022 problem 5 day 2

Source: Vietnam Mathematical Olympiad 2022 problem 5 day 2

3/5/2022
Consider 2 non-constant polynomials P(x),Q(x)P(x),Q(x), with nonnegative coefficients. The coefficients of P(x)P(x) is not larger than 20212021 and Q(x)Q(x) has at least one coefficient larger than 20212021. Assume that P(2022)=Q(2022)P(2022)=Q(2022) and P(x),Q(x)P(x),Q(x) has a root pq≠0(p,q∈Z,(p,q)=1)\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1). Prove that ∣p∣+n∣q∣≤Q(n)−P(n)|p|+n|q|\le Q(n)-P(n) for all n=1,2,...,2021n=1,2,...,2021
vmoalgebrapolynomial