Problem 933

If Masha takes all her two-ruble and five-ruble coins, she will be short of $60+60=120$ rubles for $4+5=9$ pies. On the other hand, she will be short of 60 rubles for 6 pies. That is, three pies cost 60 rubles.

## Answer

!

Difficulty: 3

Classes: $8,9,10$

Author: Frankin B.R.

In a tournament, 100 wrestlers, all of different strengths, participate. The stronger one always defeats the weaker one. The wrestlers split into pairs and held matches. Then they split into pairs differently and held matches again. Prizes were awarded to those who won both matches. What is the smallest possible number of prize winners?

## Solution

The strongest one will definitely become a prize winner.

We will show that there can be exactly one prize winner. Number the wrestlers in increasing order of strength from 1 to 100. In the first round, we will hold matches

$1-2,3-4, \ldots, 99-100$, in the second round - $100-1,2-3, \ldots, 98-99$. Then everyone, except the strongest, will lose in one of the rounds.

## Answer

One.