1. Determine the real number $a$ such that the polynomials $x^{2}+a x+1$ and $x^{2}+x+a$ have at least one common root.
(3rd Canadian Olympiad)
1. Let $x$ be a common root, then $x^{2} + a x + 1 = 0$, (1) $x^{2} + x + a = 0$, (2) Subtracting (2) from (1) gives $(a-1) \cdot (x-1) = 0$. If $a=1$, then equations (1) and (2) are identical, in which case the two polynomials have a common root. If $a \neq 1$, then the common root $x=1$, substituting into (1), (2) both yield $a=-2$. Therefore, the required $a=1$ or -2.