9.5. We will call a palindrome a natural number whose decimal representation reads the same from left to right as from right to left (the decimal representation cannot start with zero; for example, the number 1221 is a palindrome, while the numbers 1231, 1212, and 1010 are not). Which palindromes among the numbers from 10000 to 999999 are more numerous - those with an odd sum of digits or those with an even sum, and by what factor?
Answer. There are three times more palindromes with an even sum of digits.

Solution. Note that a 5-digit palindrome looks like $\overline{a b c b a}$, and a 6-digit palindrome looks like $\overline{a b c c b a}$. Each of these palindromes is uniquely determined by the first three digits $\overline{a b c}$. This means that there are as many 5-digit palindromes as there are 6-digit palindromes, which is the same as the number of numbers $\overline{a b c}$ from 100 to 999 (i.e., 900).

Note that any 6-digit palindrome has an even sum of digits $2(a+b+c)$. The sum of the digits of a 5-digit palindrome is $2(a+b)+c$, which depends only on the parity of the digit $c$. Therefore, for any fixed $a$ and $b$, there are five (even) digits $c$ for which $2(a+b)+c$ is even, and five (odd) digits $c$ for which $2(a+b)+c$ is odd. This means that there are as many 5-digit palindromes with an odd sum of digits as there are 5-digit palindromes with an even sum of digits (450 each). Therefore, there are twice as many 6-digit palindromes as there are 5-digit palindromes with an odd sum of digits. Thus, there are more palindromes from 10000 to 999999 with an even sum of digits than with an odd sum, and exactly three times more.

Remark. To prove that there are more palindromes with an even sum of digits, one can establish a one-to-one correspondence between 5-digit and 6-digit palindromes. For example, this can be done by associating each 5-digit palindrome $\overline{a b c b a}$ with a 6-digit palindrome $\overline{a b c c b a}$. In this case, the sum of the digits of all 6-digit palindromes is even, while among 5-digit palindromes, there are those with an odd sum of digits.

Comment. It is proven that there are as many 5-digit palindromes as there are 6-digit palindromes - 2 points.

It is proven that the sum of the digits of 6-digit palindromes is even - 1 point.

It is proven that the number of 5-digit palindromes with an odd sum of digits is equal to the number of 5-digit palindromes with an even sum of digits - 2 points.

It is proven that there are more palindromes with an even sum of digits, but not how many times more - 1 point.

In the case of a solution using a one-to-one correspondence, it is shown that there are more palindromes with an even sum of digits - 4 points.