Competition problem — Algebra — Olympiad Maths Prep

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Algebra Difficulty 5.7 AIME, harder Prove it Romania

If $A$ and $B$ are positive integers, then $\overline{AB}$ will denote the number obtained by writing, in order, the digits of $B$ after the digits of $A$ . For instance, if $A = 193$ and $B = 2016$ , then $\overline{AB} = 1932016$ .
Prove that there are infinitely many perfect squares of the form $\overline{AB}$ in each of the following situations:
a) $A$ and $B$ are perfect squares;
b) $A$ and $B$ are perfect cubes;
c) $A$ is a perfect cube and $B$ is a perfect square;
d) $A$ is a perfect square and $B$ is a perfect cube.

Solution

a) $A = 4$ and $B = 9$ yields $\overline{AB} = 49 = 7^2$ . Adding an even number of zeroes we get infinitely many solutions: for $A = 4$ and $B = 9 \cdot 10^{2n},\ n \in \mathbb{N}$ we get $\overline{AB} = (7 \cdot 10^n)^2$ .

b) If $A = 8$ and $B = 10^{6n},\ n \in \mathbb{N}$ , then $\overline{AB} = (9 \cdot 10^{3n})^2$ .

c) If $A = 8$ and $B = 10^{2n},\ n \in \mathbb{N}$ , then $\overline{AB} = (9 \cdot 10^n)^2$ .

d) If $A = 36$ and $B = 10^{6n},\ n \in \mathbb{N}$ , then $\overline{AB} = (19 \cdot 10^{3n})^2$ .

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