5.5. The hedgehogs collected 65 mushrooms and divided them so that each hedgehog got at least one mushroom, but no two hedgehogs had the same number of mushrooms. What is the maximum number of hedgehogs that could be
Answer: 10

Solution: If there were 11 hedgehogs, then together they would have collected no less than $1+2+3+\ldots+10+11=66$ mushrooms, which exceeds the total number of mushrooms collected. Therefore, there were fewer than 11 hedgehogs. We will show that there could have been 10 hedgehogs. Suppose the first found 1 mushroom, the second found 2 mushrooms, the third found 3, and so on, the ninth found 9 mushrooms. Then the nine hedgehogs collected $1+2+\ldots+8+9=45$ mushrooms. If the tenth found 20, then all together they collected 65 mushrooms.

Comment. Only the correct answer - 0 points. If an example for 10 hedgehogs is shown without proving the impossibility for 11, then - 4 points.