9.1 The sum of 100 numbers is 1000. The largest of these numbers was doubled, and some other number was decreased by 10. After these actions, the sum of all the numbers did not change. Find the smallest of the original numbers.
9.1 The sum of 100 numbers is 1000. The largest of these numbers was doubled, and some other number was decreased by 10. After these actions, the sum of all the numbers did not change. Find the smallest of the original numbers.

Otvet: 10.

$\boldsymbol{P e s h e n i e : ~ P u s t ь ~} M$ - the largest number, $t$ - some other number. According to the condition $2 M+t-10=M+t$, from which $M=10$. All other numbers are not greater than 10, and if at least one of them is less than 10, then the sum of all 100 numbers will be less than $100 \cdot 10=1000$. Therefore, all the numbers out of these 100 are equal to 10.