## Task B-1.1.

If $A=456+457+458+\ldots+554+555$ and $B=0.04^{3} \cdot \frac{1}{25^{2}} \cdot 125^{4}$, what is $\frac{A}{B}$?
## Solution.

By applying Gauss's trick, we calculate the sum of 100 consecutive numbers from 456 to 555:

$$
\begin{aligned}
A & =456+457+458+\ldots+554+555 \\
& =(456+555)+(457+554)+\ldots+(505+506)=50 \cdot 1011=50550
\end{aligned}
$$

By reducing to powers with base 5, we calculate the value of expression $B$.

$B=0.04^{3} \cdot \frac{1}{25^{2}} \cdot 125^{4}=\left(5^{-2}\right)^{3} \cdot \frac{1}{\left(5^{2}\right)^{2}} \cdot\left(5^{3}\right)^{4}=5^{-6} \cdot 5^{-4} \cdot 5^{12}=5^{2}=25$

Finally, the desired value is: $\frac{A}{B}=\frac{50550}{25}=2022$.

1 point

Note: The value of expression $A$ can be calculated by the student using the formula for the sum of the first $n$ natural numbers and can earn full points without proving that formula:

$$
\begin{aligned}
A & =(1+2+\ldots+555)-(1+2+\ldots+455) \\
& =\frac{555 \cdot 556}{2}-\frac{455 \cdot 456}{2}=154290-103740=50550 .
\end{aligned}
$$