Who wins : Calvin or Hobbes?
Written on a blackboard is the polynomial Calvin and Hobbes take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or decrease the coefficient of x by 1. And during his turn, Hobbes should either increase or decrease the constant coefficient by 1. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots.Who do you think has a winning strategy?
(The question is a modified form of one which appeared in INMO 2014.)
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1 solution
For i ≥ 0, let fi(x) denote the polynomial on the blackboard after Hobbes’ i-th turn. We let Calvin decrease the coefficient of x by 1. Therefore fi+1(2) = fi(2) − 1 or fi+1(2) = fi(2) − 3 (depending on whether Hobbes increases or decreases the constant term). So for some i, we have 0 ≤ fi(2) ≤ 2. If fi(2) = 0 then Calvin has won the game. If fi(2) = 2 then Calvin wins the game by reducing the coefficient of x by 1. If fi(2) = 1 then fi+1(2) = 0 or fi+1(2) = −2. In the former case, Calvin has won the game and in the latter case Calvin wins the game by increasing the coefficient of x by 1.
NOTE : In fi(x), i is in the subscript. Also, in fi+1(x) here i+1 is in the subscript.