## 1. Solve the equation

$$
x^{1996}-1996 x^{1995}+\ldots .+1=0
$$

(the coefficients of $x, x^{2}, \ldots, x^{1994}$ are unknown), given that its roots are positive real numbers.
Solution. Let $x_{1}, x_{2}, \ldots, x_{1996}$ be the solutions of the given equation. From Vieta's formulas, we have:

$$
\begin{aligned}
& x_{1}+x_{2}+\ldots+x_{1996}=1996 \\
& x_{1} \cdot x_{2} \cdot \ldots \cdot x_{1996}=1
\end{aligned}
$$

Thus,

$$
\frac{x_{1}+x_{2}+\ldots+x_{1996}}{1996}=1=1996 \sqrt{x_{1} \cdot x_{2} \cdot \ldots \cdot x_{1996}}
$$

which means the arithmetic mean is equal to the geometric mean, hence

$$
x_{1}=x_{2}=\ldots=x_{1996}=1
$$