MathDB

1

Part of 1998 Vietnam Team Selection Test

Problems(2)

| f(x) - P(x)| <= c * x^{1998}

Source: Vietnam TST 1998 for the 39th IMO, problem 1

6/26/2005
Let f(x)f(x) be a real function such that for each positive real cc there exist a polynomial P(x)P(x) (maybe dependent on cc) such that ∣f(x)−P(x)∣≤c⋅x1998| f(x) - P(x)| \leq c \cdot x^{1998} for all real xx. Prove that ff is a real polynomial.
functionalgebrapolynomiallimitalgebra unsolved
p(a) is a positive integer

Source: Vietnam TST 1998 for the 39th IMO, problem 4

6/26/2005
Find all integer polynomials P(x)P(x), the highest coefficent is 1 such that: there exist infinitely irrational numbers aa such that p(a)p(a) is a positive integer.
algebrapolynomialcalculusintegrationRational Root Theoremnumber theory unsolvednumber theory