## Subject II. (20 points)

Determine the first and last five digits of the number:

$$
\begin{array}{r}
2^{2014} \cdot 25^{1006} + 2^{2014} : \left[2^{1000} \cdot 2^{1012} + \left(2^{200} \cdot 2^{205}\right)^{5} : \left(2^{7}\right)^{2} + \left(5^{2014} : 5^{2013} - 1^{2014}\right)^{1005} \cdot 2\right] \cdot 1007 \\
\text{Prof. Ioana Ludusan, Theoretical High School "Gh. Sincai" Cluj-Napoca}
\end{array}
$$
## Subject II.

$$
\begin{aligned}
& 2^{2012} \cdot 5^{2012} \cdot 2^{2}+2^{2014}:\left[2^{2012}+\left(2^{405}\right)^{5}: 2^{14}+\left(2^{2}\right)^{1005} \cdot 2\right] \cdot 1007=10^{2012} \cdot 4+2^{2014}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_5d069eb2bbe2e614196cg-2.jpg?height=112&width=1618&top_left_y=890&top_left_x=340)

$$
\begin{aligned}
& =\underbrace{40000 \ldots 00000}_{d \theta 2012 \text { times }}+2 \cdot 1007=\underbrace{400000 \ldots 000002014}_{d z 2008 \text { times }}
\end{aligned}
$$

The first 5 digits are $4,0,0,0,0$ and the last five digits are $0,2,0,1,4$.