Home > Olympiad > IMC 2012 Problem 6

## IMC 2012 Problem 6

Consider a polynomial

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

Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients ${a_0..a_{2011}}$ and assign a real value to it. Albert has the first move. Once a value has been assigned to a coefficient it cannot be changed anymore. The game ends after all the coefficients have been assigned values.

Homer’s goal is to make the polynomial ${f(x)}$ divisible by a given 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 ${m(x)=x^2+1}$?