1971 Canadian MO Problems/Problem 4

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Problem

Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.

Solution

Let this root be $r$ . Then we have

$\begin{matrix} r^2 + ar + 1 &=& r^2 + r + a\\ar + 1 &=& r + a\\(a-1)r &=& (a-1)\end{matrix}$

Now, if $a = 1$ , then we're done, since this satisfies the problem's conditions. If $a \neq 1$ , then we can divide both sides by $(a - 1)$ to obtain $r = 1$ . Substituting this value into the first polynomial gives

$\begin{matrix} 1 + a + 1 &=& 0\\a &=& -2 \end{matrix}$

It is easy to see that this value works for the second polynomial as well.

Therefore the only possible values of $a$ are $1$ and $-2$ . Q.E.D.

1971 Canadian MO (Problems)
Preceded by Problem 3	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10	Followed by Problem 5

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1971 Canadian MO Problems/Problem 4

From AoPSWiki

Problem

Solution