7. The numbers $a, b$, and $c$ were written on the board. They were erased, and in their place, the numbers $a-1, b+1, c^{2}$ were written. It turned out that the numbers on the board were the same as those initially written (possibly in a different order). What values can the number $a$ take, given that the sum of the numbers $a, b$, and $c$ is 2005? If necessary, round the answer to the hundredths.
Answer: 1003 or 1002.5.

Solution. The sum of all numbers does not change, so $a+b+c=(a-1)+(b+1)+c^{2}$. From this, $c^{2}=c$, hence $c=1$ or $c=0$. In both cases, the number $c$ equals the number $c^{2}$, then the number $a$ will be equal to the number $b+1$ (and the number $b$ will be equal to the number $a-1$).

If $c=1$, then $a+b=2004$. From this, $(b+1)+b=2004, b=(2004-1) / 2=1001.5$ and $a=b+1=1002.5$.

If $c=0$, then $a+b=2005$. From this, $(b+1)+b=2005, b=(2005-1) / 2=1002$ and $a=b+1=1003$.