Squares Ending in 5 Occurrences of 2-Digit Pattern

Theorem
Let $n$ be a square number whose decimal representation ends in the pattern $\mathtt {xyxyxyxyxyxy}$.

Then $\mathtt {xy}$ is one of:
 * $21, 29, 61, 69, 84$

The smallest example of such a number is:
 * $508 \, 853 \, 989^2 = 258 \, 932 \, 382 \, 121 \, 212 \, 121$