MathDB

4

Part of 2008 Moldova Team Selection Test

Problems(3)

Probably Known Property of homogeneous Polynomials in R[X,Y]

Source: Moldova 2008 IMO-BMO First TST Problem 4

3/3/2008
A non-zero polynomial S∈R[X,Y] S\in\mathbb{R}[X,Y] is called homogeneous of degree d d if there is a positive integer d d so that S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y) for any λ∈R \lambda\in\mathbb{R}. Let P,Q∈R[X,Y] P,Q\in\mathbb{R}[X,Y] so that Q Q is homogeneous and P P divides Q Q (that is, P∣Q P|Q). Prove that P P is homogeneous too.
algebrapolynomialfactorialcalculusintegrationalgebra proposed
Nice: find number of even permutations with no fixed points

Source: Moldova 2008 IMO-BMO Second TST Problem 4

3/29/2008
Find the number of even permutations of {1,2,…,n} \{1,2,\ldots,n\} with no fixed points.
countingderangementinductionlinear algebracombinatorics proposedcombinatorics
Good sets and coloring of all positive integers.

Source: Moldova 2008 IMO-BMO Third TST Problem 4

3/30/2008
A non-empty set S S of positive integers is said to be good if there is a coloring with 2008 2008 colors of all positive integers so that no number in S S is the sum of two different positive integers (not necessarily in S S) of the same color. Find the largest value t t can take so that the set S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\} is good, for any positive integer a a. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...
searchfloor functioncombinatorics proposedcombinatorics