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Indian National Mathematical Olympiad

Combinatorics Difficulty 6.2 National Olympiad Prove it India

Problem:

Written on a blackboard is the polynomial $x^{2} + x + 2014$ . 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. Prove that Calvin has a winning strategy.

Solution

Solution:

For $i \geq 0$ , let $f_{i}(x)$ denote the polynomial on the blackboard after Hobbes' $i$ -th turn. We let Calvin decrease the coefficient of $x$ by $1$ . Therefore $f_{i+1}(2) = f_{i}(2) - 1$ or $f_{i+1}(2) = f_{i}(2) - 3$ (depending on whether Hobbes increases or decreases the constant term). So for some $i$ , we have $0 \leq f_{i}(2) \leq 2$ . If $f_{i}(2) = 0$ then Calvin has won the game. If $f_{i}(2) = 2$ then Calvin wins the game by reducing the coefficient of $x$ by $1$ . If $f_{i}(2) = 1$ then $f_{i+1}(2) = 0$ or $f_{i+1}(2) = -2$ . 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 $1$ .

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