3

Part of 2001 China Team Selection Test

Problems(2)

f[x^k + f(y)] = y +[f(x)]^k

Source: China TST 2001, problem 3

For a given natural number

k > 1

, find all functions

f:\mathbb{R} \to \mathbb{R}

such that for all

x, y \in \mathbb{R}

f[x^k + f(y)] = y +[f(x)]^k

functionalgebrapolynomialfunctional equationalgebra unsolved

Minimize |x^3 - ax^2 - bx - c|

Source: China TST 2001, problem 6

F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|

a

b

c

run over all the real numbers, find the smallest possible value of

F

algebrafunctioncalculuspolynomialminimization