MathDB

3

Part of 2000 Vietnam National Olympiad

Problems(2)

Vietnam NMO 2000_3

Source:

10/26/2008
Consider the polynomial P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38. (a) Prove that there are at least nine integers a a in the interval [1,32000] [1, 3^{2000}] for which P(a) P(a) is divisible by 32000 3^{2000}. (b) Find the number of integers a a in [1,32000] [1, 3^{2000}] with the property from (a).
algebrapolynomialnumber theory unsolvednumber theory
Vietnam NMO 2000_6

Source:

10/26/2008
Let P(x) P(x) be a nonzero polynomial such that, for all real numbers x x, P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x). Determine the maximum possible number of real roots of P(x) P(x).
algebrapolynomialcomplex numbersalgebra unsolved