MathDB

2

Part of 1999 China National Olympiad

Problems(2)

Results about the sequence of polynomials f_n(x)

Source: Chinese MO 1999

3/19/2012
Let aa be a real number. Let (fn(x))n≥0(f_n(x))_{n\ge 0} be a sequence of polynomials such that f0(x)=1f_0(x)=1 and fn+1(x)=xfn(x)+fn(ax)f_{n+1}(x)=xf_n(x)+f_n(ax) for all non-negative integers nn. a) Prove that fn(x)=xnfn(x−1)f_n(x)=x^nf_n\left(x^{-1}\right) for all non-negative integers nn. b) Find an explicit expression for fn(x)f_n(x).
algebrapolynomialfunctioninductionalgebra proposed
Maximum value of lambda

Source: Chinese MO 1999

3/19/2012
Determine the maximum value of λ\lambda such that if f(x)=x3+ax2+bx+cf(x) = x^3 +ax^2 +bx+c is a cubic polynomial with all its roots nonnegative, then f(x)≥λ(x−a)3f(x)\geq\lambda(x -a)^3 for all x≥0x\geq0. Find the equality condition.
algebrapolynomialinequalities proposedinequalities