MathDB

1

Part of 2017 China Northern MO

Problems(2)

China Northern Mathematical Olympiad 2017, Problem 1

Source: China Northern Mathematical Olympiad 2017

7/29/2017
A sequence {an}\{a_n\} is defined as follows: a1=1a_1 = 1, a2=13a_2 = \frac{1}{3}, and for all n≥1,n \geq 1, (1+an)(1+an+2)(1+an+1)2=anan+2an+12\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}.
Prove that, for all n≥1n \geq 1, a1+a2+...+an<3421a_1 + a_2 + ... + a_n < \frac{34}{21}.
inequalitiesSequence
2017 CNMO Grade 11 P1

Source: 2017 China Northern MO, Grade 11, Problem 1

2/24/2020
Define sequence (an):a1=e,a2=e3,e1−kank+2=an+1an−12k(a_n):a_1=\text{e},a_2=\text{e}^3,\text{e}^{1-k}a_n^{k+2}=a_{n+1}a_{n-1}^{2k} for all n≥2n\geq2, where kk is a positive real number. Find ∏i=12017ai\prod_{i=1}^{2017}a_i.
algebra