MathDB
LCM is larger than argument/result

Source: 2012 Indonesia Round 2.5 TST 2 Problem 1

May 21, 2012
number theoryleast common multiplealgebrapolynomialalgebra unsolved

Problem Statement

Given a positive integer nn.
(a) If PP is a polynomial of degree nn where P(x)∈ZP(x) \in \mathbb{Z} for every x∈Zx \in \mathbb{Z}, prove that for every a,b∈Za,b \in \mathbb{Z} where P(a)≠P(b)P(a) \neq P(b), lcm(1,2,…,n)≥∣a−bP(a)−P(b)∣\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|
(b) Find one PP (for each nn) such that the equality case above is achieved for some a,b∈Za,b \in \mathbb{Z}.
Back to ProblemsView on AoPS