5. Among the numbers from 1 to $10^{23}$, which are more numerous - those with a two-digit sum of digits or those with a three-digit sum? Answer: those with a three-digit sum.
Solution. The number $10^{23}$ has a digit sum of 1, so we will not consider it. We will write numbers as 23-digit numbers, adding leading zeros. We will call digits "complementary" if their sum is 9. To each number with a two-digit digit sum, we will correspond a number by replacing each digit with its "complementary" counterpart. Each number will be paired with a unique number. If the digit sum of a number is x, then the digit sum of its paired number will be $9 * 23 - x = 207 - x \geq 207 - 99 = 108$, and it will not exceed $9 * 23$, that is, it will be a three-digit number. Therefore, the number of numbers with a three-digit digit sum is no less than those with a two-digit digit sum. However, there is also, for example, the number $99...99$ (23 nines), which is not paired with any number.