MathDB

2

Part of 2004 China National Olympiad

Problems(2)

Expression for x_n in terms of n and x_1

Source: Chinese MO 2004

9/25/2011
Let cc be a positive integer. Consider the sequence x1,x2,…x_1,x_2,\ldots which satisfies x1=cx_1=c and, for n≥2n\ge 2, xn=xn−1+⌊2xn−1−(n+2)n⌋+1x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1 where ⌊x⌋\lfloor x\rfloor denotes the largest integer not greater than xx. Determine an expression for xnx_n in terms of nn and cc.
Huang Yumin
floor functionmodular arithmeticinductionalgebra proposedalgebra
Inequality for the n positive integers a_i

Source: Chinese MO 2004

11/3/2008
For a given positive integer n≥2n\ge 2, suppose positive integers aia_i where 1≤i≤n1\le i\le n satisfy a1<a2<…<ana_1<a_2<\ldots <a_n and ∑i=1n1ai≤1\sum_{i=1}^n \frac{1}{a_i}\le 1. Prove that, for any real number xx, the following inequality holds (∑i=1n1ai2+x2)2≤12⋅1a1(a1−1)+x2\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2}
Li Shenghong
inequalitiesalgebrapolynomialn-variable inequality