MathDB

2

Part of 1990 Vietnam National Olympiad

Problems(2)

Removing n numbers from {1,...,2n-1} to leave maximal sum

Source: Vietnam NMO 1990 question 2

10/26/2008
At least n−1 n - 1 numbers are removed from the set {1,2,…,2n−1}\{1, 2, \ldots, 2n - 1\} according to the following rules: (i) If a a is removed, so is 2a 2a; (ii) If a a and b b are removed, so is a \plus{} b. Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.
combinatorics unsolvedcombinatorics
Vietnam NMO 1990_5

Source:

10/26/2008
Suppose f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n (a0≠0 a_0\neq 0) is a polynomial with real coefficients satisfying f(x)f(2x^2) \equal{} f(2x^3 \plus{} x) for all x∈R x \in\mathbb{R}. Prove that f(x) f(x) has no real roots.
algebrapolynomialfunctional equationalgebra unsolved