Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Prove it Ukraine

Positive integers $A$ and $B$ have the following decimal forms: $A = \overline{abcabc}$ and $B = \overline{d00d}$ , where $a, b, c, d$ are decimal digits, $a \neq 0, d \neq 0$ . Find all possible values of $a, b, c, d$ such that $A+B$ is a full square.

Solution

Оскільки
$A + B = \overline{abcabc} + \overline{d00d} = 1001 (\overline{abc} + d) \le 1001 (999 + 9) = 1001 \cdot 1008,$
то для виконання умови задачі необхідно й достатньо, щоб $\overline{abc} + d = 1001$ . Звідси $a = 9, b = 9$ і $c + d = 11$ .

Відповідь: $a = 9, b = 9, c = 11 - d, a, d \in \{2, 3, 4, 5, 6, 7, 8, 9\}$ .

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