HMMT 二月 2019 · ALGNT 赛 · 第 1 题

HMMT February 2019 — ALGNT Round — Problem 1

What is the smallest positive integer that cannot be written as the sum of two nonnegative palin- dromic integers? (An integer is palindromic if the sequence of decimal digits are the same when read backwards.) 2 N x 2 N ( ) ( ) (2 )

解析

What is the smallest positive integer that cannot be written as the sum of two nonnegative palin- dromic integers? (An integer is palindromic if the sequence of decimal digits are the same when read backwards.) Proposed by: Yuan Yao Answer: 21 We need to first prove that every positive integer N less than 21 can be written as sum of two nonnegative palindromic integers. If N is in the interval [1 , 9], then it can be written as 0 + N . If N is in the interval [10 , 18], it can be written as 9 + ( N − 9). In addition, 19 and 20 can be written as 11 + 8 and 11 + 9, respectively. Second, we need to show that 21 cannot be expressed in such a way. Lets suppose 21 = a + b with a ≤ b . It follows that b has to be at least 11. Since b ≤ 21, the only way for b to be palindromic is that b = 11. However, this leads to a = 21 − b = 10, which is not a palindrome. Therefore, 21 is the smallest number that satisfy the problem condition. 2 N x 2 N ( ) ( ) (2 )