MathDB

4

Part of 2008 IMC

Problems(2)

IMC 2008 Day 1 P4 - Better Triples

Source: Problem 4

7/30/2008
We say a triple of real numbers (a1,a2,a3) (a_1,a_2,a_3) is better than another triple (b1,b2,b3) (b_1,b_2,b_3) when exactly two out of the three following inequalities hold: a1>b1 a_1 > b_1, a2>b2 a_2 > b_2, a3>b3 a_3 > b_3. We call a triple of real numbers special when they are nonnegative and their sum is 1 1. For which natural numbers n n does there exist a collection S S of special triples, with |S| \equal{} n, such that any special triple is bettered by at least one element of S S?
inequalitieslinear algebramatrixgroup theoryabstract algebraanalytic geometrygeometry
IMC 2008 Day 2 P4 - Polynomial with degree > 5

Source: Problem 4

7/28/2008
Let Z[x] \mathbb{Z}[x] be the ring of polynomials with integer coefficients, and let f(x),g(x)∈Z[x] f(x), g(x) \in\mathbb{Z}[x] be nonconstant polynomials such that g(x) g(x) divides f(x) f(x) in Z[x] \mathbb{Z}[x]. Prove that if the polynomial f(x)\minus{}2008 has at least 81 distinct integer roots, then the degree of g(x) g(x) is greater than 5.
algebrapolynomialpigeonhole principleIMCcollege contests