MathDB

2

Part of 2011 China Second Round Olympiad

Problems(2)

a polynomial

Source: 2011-2012 china second round,problem 2

10/30/2011
For any integer n≥4n\ge 4, prove that there exists a nn-degree polynomial f(x)=xn+an−1xn−1+⋯+a0f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0 satisfying the two following properties:
(1) aia_i is a positive integer for any i=0,1,…,n−1i=0,1,\ldots,n-1, and
(2) For any two positive integers mm and kk (k≥2k\ge 2) there exist distinct positive integers r1,r2,...,rkr_1,r_2,...,r_k, such that f(m)≠∏i=1kf(ri)f(m)\ne\prod_{i=1}^{k}f(r_i).
algebrapolynomialgeometrygeometric transformationnumber theory proposednumber theory
Range of a function

Source: China second round Test 1

1/30/2012
Find the range of the function f(x)=x2+1x−1f(x)=\frac{\sqrt{x^2+1}}{x-1}.
functionalgebra unsolvedalgebra