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India — Team Selection Test, 2013

Number theory Difficulty 5.4 AIME, harder Prove it India

A positive integer $a$ is called a double number if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2\cdots a_k a_1a_2\cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$ , and $a_1 \ne 0$ . For example, 283283 is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of 10.

Solution

The answer is affirmative. Let $k \ge 0$ be such that $k \equiv 15 \pmod{42}$ and $b = 5(10^k + 1)/7 + 1$ (which is an integer). Then $c = \frac{5}{7}(b+1)$ is an integer and $b^2 - 1 = (10^k + 1)c$ is a double number.

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