# Problem 2.

For each integer value of the parameter $K$, solve the system of equations

$$
\left\{\begin{aligned}
2[x]+y & =3 / 2, \\
([x]-x)^{2}-2[y] & =K
\end{aligned}\right.
$$

Here $[x]$ denotes the integer part of the number $x$.
# Solution.

Let

$$
x=M+a, \quad y=N+b, \quad M, N \in \mathbb{Z}, \quad a, b \in[0 ; 1)
$$

From the first equation, we get $2 M+N+b=3 / 2$ and then

$$
b=0.5, \quad N=1-2 M
$$

Substitute these values into the second equation:

$$
(-a)^{2}-2(1-2 M)=K
$$

Then

$$
a=0, \quad K=4 M-2, \quad x=M, \quad y=3 / 2-2 M
$$

## Answer.

If $K=4 M-2$, where $M \in \mathbb{Z}$, then $x=M, y=3 / 2-2 M$.

For other $K$, there are no solutions.