# Problem 5. Maximum 15 points

The great-grandfather-banker left a legacy to his newborn great-grandson. According to the agreement with the bank, the amount in the great-grandson's account increases. Every year, on the day after the birthday, the current amount is increased by 1 million rubles more than in the previous year. Thus, if the initial amount was zero rubles, after one year +1 million rubles, after 2 years $1+2$ million, after 3 years $1+2+3$ and so on. According to the terms of the agreement, the process will stop, and the great-grandson will receive the money when the amount in the account becomes a three-digit number consisting of three identical digits.

How old will the great-grandson be when the terms of the agreement are met?
# Solution:

Since the number consists of identical digits, it can be represented as 111 multiplied by a. According to the problem, the same number should be obtained as the sum of an arithmetic progression. The first element of the progression is 1, the last is \( \mathrm{n} \), and the number of elements in the progression is \( \mathrm{n} \). Since \( 111 = 3 \times 37 \):

\[
n(n+1) = 2 \times 3 \times 37 \times a
\]

Since 37 is a prime number, and \( \frac{n(n+1)}{2} \) should be less than 1000, then either \( \mathrm{n}+1=37 \) or \( \mathrm{n}=37 \). By checking, we find that \( \mathrm{n}=36 \), and the required three-digit number consisting of identical digits is 666.

## Criteria:

Correctly formulated equation in integers: 5 points;

Correct answer: 5 points;

Analysis of the equation and proof of the absence of other options: 5 points.