MathDB

2

Part of 2012 Pre - Vietnam Mathematical Olympiad

Problems(2)

Compute limit

Source: Pre-VMO 2012 - Round 2 - Problem 2

12/25/2011
Compute lim⁡n→∞{(2+3)n}\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}
limittopologyalgebra proposedalgebra
Determine the first term of a sequence

Source: Pre-VMO 2012 - Round 2 - Problem 6

12/26/2011
Let (an)(a_n) defined by: a0=1,  a1=p,  a2=p(p−1)a_0=1, \; a_1=p, \; a_2=p(p-1), an+3=pan+2−pan+1+an,  ∀n∈Na_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}. Knowing that
(i) an>0,  ∀n∈Na_n>0, \; \forall n \in \mathbb{N}.
(ii) aman>am+1an−1,  ∀m≥n≥0a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0.
Prove that ∣p−1∣≥2|p-1| \ge 2.
algebrapolynomialalgebra proposed