$a_1a_2a_3$ and $a_3a_2a_1$ are two three-digit decimal numbers, with $a_1$ and $a_3$ different non-zero digits. Squares of these numbers are five-digit numbers $b_1b_2b_3b_4b_5$ and $b_5b_4b_3b_2b_1$ respectively. Find all such three-digit numbers.
1. **Identify the range of the three-digit numbers:**
   - Given that $a_1a_2a_3$ and $a_3a_2a_1$ are three-digit numbers, their squares are five-digit numbers. Therefore, we have:
     \[
     10000 \leq (a_1a_2a_3)^2 \leq 99999
     \]
   - Taking the square root of the bounds:
     \[
     100 \leq a_1a_2a_3 \leq 316
     \]
   - Thus, $a_1a_2a_3$ and $a_3a_2a_1$ must be in the range $[100, 316]$.

2. **Determine possible values for $a_1$ and $a_3$:**
   - Since $a_1$ and $a_3$ are non-zero digits and must be different, they can be any digits from 1 to 9, but not equal to each other.
   - We will consider the cases where $a_1 < a_3$ to avoid redundancy.

3. **Case 1: $(a_1, a_3) = (1, 2)$**
   - The numbers are $\overline{1a2}$ and $\overline{2a1}$.
   - Their squares must be five-digit numbers and palindromes.
   - The maximum value for these numbers is less than 224 (since $224^2 = 50176$ which is not a palindrome).
   - Possible values for $a$ are 0, 1, and 2:
     - $102^2 = 10404$ (palindrome)
     - $112^2 = 12544$ (not a palindrome)
     - $122^2 = 14884$ (palindrome)

4. **Case 2: $(a_1, a_3) = (1, 3)$**
   - The numbers are $\overline{1a3}$ and $\overline{3a1}$.
   - Their squares must be five-digit numbers and palindromes.
   - The maximum value for these numbers is less than 317 (since $317^2 = 100489$ which is not a five-digit number).
   - Possible values for $a$ are 0 and 1:
     - $103^2 = 10609$ (palindrome)
     - $113^2 = 12769$ (not a palindrome)

5. **Case 3: $(a_1, a_3) = (2, 3)$**
   - The numbers are $\overline{2a3}$ and $\overline{3a2}$.
   - Their squares must be five-digit numbers and palindromes.
   - The maximum value for these numbers is less than 317.
   - Possible values for $a$ are 0 and 1:
     - $203^2 = 41209$ (not a palindrome)
     - $213^2 = 45369$ (not a palindrome)

6. **Conclusion:**
   - The valid three-digit numbers whose squares are five-digit palindromes are:
     \[
     \boxed{102, 103, 122}
     \]