Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Prove it Soviet Union

Problem:

Starting with the polynomial $x^2 + 10x + 20$ , a move is to change the coefficient of $x$ by $1$ or to change the coefficient of $x^0$ by $1$ (but not both). After a series of moves the polynomial is changed to $x^2 + 20x + 10$ . Is it true that at some intermediate point the polynomial had integer roots?

Solution

Solution:

We have $x^2 + (n + 1)x + n = (x + n)(x + 1)$ , so $x^2 + a x + b$ has integer roots if $a = b + 1$ (and $a$ and $b$ are integers). But initially $a - b$ is $-10$ and it ends up as $+10$ . Each move changes $a - b$ by $\pm 1$ , so it must pass through all values between $-10$ and $+10$ .

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