1. (2 points) Boy Vasya wrote down the non-zero coefficients of the polynomial $P(x)$ of the tenth degree in his notebook. Then he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, and so on, until he obtained a constant, which he also wrote down.

What is the smallest number of different numbers he could have obtained?

Coefficients are written with their sign, constant terms are also written if there is a monomial of the form $\pm x^{n}$, $\pm 1$ is written.
Answer: 10

Solution: Evaluation: since the polynomial has a degree of 10, it definitely has a non-zero coefficient for $x^{10}$, let's call it $a$. Then the leading coefficient of the derivative of this polynomial is $10a$, the leading coefficient of the second derivative is $10 \cdot 9a$, and so on, the leading coefficients of the ninth and tenth derivatives are $10!a$, and all these numbers, except for the last two, are distinct, thus there are definitely 10 distinct numbers.

Example: $\frac{x^{10}}{10!}+\frac{x^{9}}{9!}+\ldots+x+1$ gives us exactly 10 distinct numbers, since each subsequent term is the derivative of the previous one.

Also, any polynomial proportional to this one, or differing from a proportional one by omitting some terms, such as $x^{10}$, will work. It is easy to prove that there are no other examples, although this is not required in the problem.