[ $[$ Invariants $]$

On the board, the numbers $1,2,3, \ldots, 19,20$ are written. It is allowed to erase any two numbers $a$ and $b$ and write the number $a+b-1$ instead.

What number can remain on the board after 19 such operations?

#
For any set of $n$ numbers on the board, consider the following quantity $X$: the sum of all numbers, decreased by $n$. It is not hard to verify that this is an invariant. In the set from the condition, $X=(1+2+\ldots+20)-20=190$. After 19 operations, when there is one number $p$ left on the board, $X=p-1$. Therefore, $p=191$.

## Answer

191.