HMMT February, 2018

Algebra Difficulty 5.3 AIME, harder Prove it United States

Problem:

Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a < b$ and there are no other five-digit palindromes strictly between $a$ and $b$ . What are all possible values of $b - a$ ? (A number is a palindrome if it reads the same forwards and backwards in base 10.)

Solution

Solution:

Let $\overline{x y z y x}$ be the digits of the palindrome $a$ . There are three cases.

If $z < 9$ , then the next palindrome greater than $\overline{x y z y x}$ is $\overline{x y (z+1) y x}$ , which differs by $100$ .

If $z = 9$ but $y < 9$ , then the next palindrome up is $\overline{x (y+1) 0 (y+1) x}$ , which differs from $\overline{x y 9 y x}$ by $110$ .

Finally, if $y = z = 9$ , then the next palindrome after $\overline{x 9 9 9 x}$ is $\overline{(x+1) 0 0 0 (x+1)}$ , which gives a difference of $11$ .

Thus, the possible differences are $11, 100, 110$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.