The board had an equation of the form $x^{2}+p x+q=0$ with integer non-zero coefficients $p$ and $q$.

From time to time, different schoolchildren approached the board, erased the equation, and then composed and wrote down an equation of the same form, the roots of which were the coefficients of the erased equation. At some point, the composed equation matched the one that was originally written on the board. What was the original equation written on the board?
According to Vieta's theorem, the coefficients of the new quadratic polynomial are $-(p+q)$ and $pq$ respectively. Note that the second coefficient (of $x$) of none of the written quadratic polynomials can be zero. Indeed, in this case, the constant term of all subsequent quadratic polynomials would be zero, which means that the original polynomial could not be among them. Further, for each subsequent quadratic polynomial, the absolute value of the constant term is not less than that of the previous one. Therefore, the absolute values of all constant terms are equal, which means that all second coefficients are $\pm 1$. Thus, $-(p+q) = \pm 1, p = \pm 1$. Therefore, for the first quadratic polynomial, there are only two possible options: $x^{2}+x-2$ and $x^{2}-x+2$. It is easy to check that in the first case, the quadratic polynomial does not change, while in the second case, it first becomes the quadratic polynomial $x^{2}-x-2$, and then the quadratic polynomial $x^{2}+3x+2$, the absolute value of the second coefficient of which is not equal to one.

## Answer

$x^{2}+x-2$

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