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Number theory Difficulty 3.0 AMC 10/12 Find the answer United Kingdom

Each of the positive integers $a$ , $b$ , and $c$ has three digits, and for each of these integers the first digit is the same as its last digit. Also $b = 2a + 1$ and $c = 2b + 1$ . How many possibilities are there for the integer $a$ ?

(A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) more than $3$

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