HMMT February, 2019

Solution:

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. Let's suppose $21=a+b$ with $a \leq b$. It follows that $b$ has to be at least $11$. Since $b \leq 21$, 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 satisfies the problem condition.