MathDB

Consider a polynomial

f(x)=x^{2012}+a_{2011}x^{2011}+\dots+a_1x+a_0.

Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients

a_0,a_1,\dots,a_{2011}

and assign a real value to it. Albert has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values. Homer's goal is to make

f(x)

divisible by a fixed polynomial

m(x)

and Albert's goal is to prevent this. (a) Which of the players has a winning strategy if

m(x)=x-2012

? (b) Which of the players has a winning strategy if

m(x)=x^2+1

?
Proposed by Fedor Duzhin, Nanyang Technological University.

Problem Statement