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Finite algebraic moves on a blackboard

Source: 2021 Turkey JBMO TST P7

May 24, 2021
algebraalgebra proposed

Problem Statement

Initially on a blackboard, the equation a1x2+b1x+c=0a_1x^2+b_1x+c=0 is written where a1,b1,c1a_1, b_1, c_1 are integers and (a1+c1)b1>0(a_1+c_1)b_1 > 0. At each move, if the equation ax2+bx+c=0ax^2+bx+c=0 is written on the board and there is a x∈Rx \in \mathbb{R} satisfying the equation, Alice turns this equation into (b+c)x2+(c+a)x+(a+b)=0(b+c)x^2+(c+a)x+(a+b)=0. Prove that Alice will stop after a finite number of moves.
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