3. Today is March 24, 2007. The sum of the digits of the year is 9. After how many years will the same sum of digits repeat? How many times will the same sum occur until the year 2107 inclusive?
3. In the record of the year, the first digit does not change, so the sum of the second, third, and fourth digits should be equal to 7 $(9-2=7)$. Since the largest year is 2107, we need to consider numbers of the form $20 \mathrm{XX}$ and $210 \mathrm{X}$. Let's consider each of these cases separately.

Let's list all numbers of the first form $-20 \mathrm{XX}$ that satisfy the condition of the problem:

2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070.

The year closest to 2007 is 2016. Therefore, the sum of the digits will repeat in 9 years $(2016-2007=9)$. Let's list the numbers of the second form $-210 X: 2106$.

In total, up to the year 2107, this sum will repeat 8 times.

Answer: the smallest number of years is 9, and the sum will repeat 8 times.