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}?
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