Brazilian Math Olympiad

a. First notice that $86$ is pal. Then it's not hard to check by hand that every number from $87$ to $99$ is not pal.

b. The answer is yes. First consider the $2011$-digit number $11\dots 1$. The sum of its digits is $2011$. The smallest perfect square greater than $2011$ is $45^2=2025$. Since $2025-2011=14$ and $14=2\cdot (2^2-1^2)+(3^2-1^2)$, we can exchange two $1$s by two $2$s and one $1$ by one $3$. So we obtain the pal number $11\dots 123$.