Petr brought a bag of candies to ten friends and distributed them so that each got the same amount. Then he realized that the bag had the smallest possible number of candies that could also be distributed in such a way that each of the friends would get a different (but non-zero) number.

Determine how many candies were originally in the bag, and provide an example of the second mentioned distribution method.

(T. Bárta)
According to how Petr distributed the candies, we know that their total number was divisible by ten. To be able to distribute the candies in the second way, he would need at least

$$
1+2+3+4+5+6+7+8+9+10=55 .
$$

The smallest number satisfying both conditions is 60. This number can be distributed in the second way, for example, by giving nine friends $1,2,3,4,5,6,7,8$, 9 and the remaining 15 candies to the last one.

There were 60 candies in the bag.

Evaluation. 3 points for partial insights; 3 points for an example of the second distribution and the conclusion.