$\underline{\text { Chernyaev N.L. }}$

Ten children were given 100 spaghetti noodles each in their plates. The children did not want to eat and started playing. In one move, one of the children transfers one noodle from their plate to each of the other children. After what minimum number of moves can all the children have a different number of noodles in their plates?
Evaluation. Consider the difference between the number of macaroni pieces Petya has and the number of macaroni pieces Vasya has. Petya's action will reduce this difference by 10, Vasya's action will increase it by 10, and the actions of other children will not change it. Therefore, this difference will not be zero if and only if Petya and Vasya perform a different number of actions.

Thus, for all children to end up with a different number of macaroni pieces, they need to perform at least $0+$ $1+2+\ldots+9=45$ actions.

Example. The first child will perform 0 actions, the second - 1, the third - 2, ..., the tenth - 9. Each will give away no more than $9 \cdot 9=81$ macaroni pieces, and the initial 100 macaroni pieces are sufficient for this.

## Answer

After 45 actions.