2. (7p) Consider the numbers:

$$
\begin{aligned}
& A=0^{1}+1^{0}+2^{3}+3^{2}+4^{5}+5^{4}+349 \\
& B=2^{2000} \cdot\left[3^{10}: 3^{7}-\left(5^{4}\right)^{5}: 25^{9}\right]^{15}+16^{250} \cdot 8^{250} \cdot 4^{250}: 2^{235} \\
& C=1+2+2^{2}+2^{3}+\ldots .+2^{2015}
\end{aligned}
$$

Determine which of the numbers $A, B$, and $C$ are perfect squares.

Felicia Brodețchi
2. $A=2016$ lies between $44^{2}$ and $45^{2}$, so it is not a perfect square ..... (2p)
$B=2^{2015}+2^{2015}=2^{2016}=\left(2^{1008}\right)^{2}$ is a perfect square ..... (2p)
$C=2^{2016}-1$ ..... (2p)
$C$ is the predecessor of $\left(2^{1008}\right)^{2}$, so it is not a perfect square ..... (1p)