3. A natural number $n$ is called lucky if it can be uniquely decomposed into the sum of 10 different natural numbers (the order of the summands does not matter). Find all lucky numbers.

$(25$ points. $)$
Answer: 55 and 56.

Let the number $n$ be lucky, $n=a_{1}+a_{2}+\ldots+a_{10}$, where $a_{1}1$, then $n$ can be split into a sum of different natural addends in another way: $n=\left(a_{1}-1\right)+a_{2}+\ldots+\left(a_{10}+1\right)$. Thus, $a_{1}=1$.

Next, if we assume that $a_{2}>2$, then for $n$ we can again provide another partition: $n=a_{1}+\left(a_{2}-1\right)+a_{3}+\ldots+\left(a_{10}+1\right)$. Therefore, $a_{2}=2$. Continuing in this manner, we get $a_{3}=3$, $a_{4}=4, \ldots, a_{9}=9$. If $a_{10}>11$, then $a_{9}+1<a_{10}-1$, and we can again construct another partition.

Finally, it is not difficult to see that for $a_{10}=10$ or $a_{10}=11$, the resulting numbers 55 and 56 are lucky.

Criteria. 5 points for each correct example of a lucky number. Proving that there are no other numbers - another 15 points.