HMMT 二月 2018 · COMB 赛 · 第 2 题

HMMT February 2018 — COMB Round — Problem 2

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.)

解析

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.) Proposed by: Kevin Sun Answer: 100 , 110 , 11 Let xyzyx be the digits of the palindrome a . There are three cases. If z < 9, then the next palindrome greater than xyzyx is xy ( z + 1) yx , which differs by 100. If z = 9 but y < 9, then the next palindrome up is x ( y + 1)0( y + 1) x , which differs from xy 9 yx by 110. Finally, if y = z = 9, then the next palindrome after x 999 x is ( x + 1)000( x + 1), which gives a difference of 11. Thus, the possible differences are 11 , 100 , 110.