Auto: Zasavsky A.A.

The analyst made a forecast of the dollar exchange rate change for each of the 12 upcoming months: by what percentage (a number greater than $0 \%$ and less than $100 \%$) the rate would change in October, by what percentage in November, ..., and by what percentage in September. It turned out that for each month, he correctly predicted the percentage change in the rate, but was wrong about the direction of the change (that is, if he predicted that the rate would increase by $x \%$, then the rate fell by $x \%$, and vice versa). At the same time, after 12 months, the rate matched the forecast. In which direction did the rate ultimately change? #
We will use fractions instead of percentages: the exchange rate changes by $x(00)$. Then, by the condition,

$a=\left(1+x_{1}\right)\left(1+x_{2}\right) \ldots\left(1+x_{12}\right)=\left(1-x_{1}\right)\left(1-x_{2}\right) \ldots\left(1-x_{12}\right)$, from which $a^{2}=\left(1-x_{1}^{2}\right)\left(1-x_{2}^{2}\right) \ldots\left(1-x_{12}^{2}\right)<1$, and, consequently, $a<1$.

## Answer

It decreased.