3.1. Kolya had 10 sheets of paper. On the first step, he chooses one sheet and divides it into two parts. On the second step, he chooses one sheet from the available ones and divides it into 3 parts, on the third step, he chooses one sheet from the available ones and divides it into 4, and so on. After which step will the number of sheets first exceed 500?
# Answer: 31

Solution. On the $k$-th step, the number of leaves increases by $k$, so after $k$ steps, the number of leaves will be $10+(1+2+\ldots+k)=10+\frac{k(k+1)}{2}$. When $k=30$, this number is less than 500 (it equals 475), and when $k=31$, it is already more than 500 (it equals 506), so the answer to the problem is 31.