2018 AMC 10A Problem 25

Let $R = \dfrac{10^n - 1}{9} = 11\ldots1$ ($n$ ones). Then $A_n = aR$, $B_n = bR$, and, since $10^{2n} - 1 = (10^n - 1)(10^n + 1)$ with $10^n + 1 = 9R + 2$,
$$C_n = c \cdot \frac{10^{2n} - 1}{9} = cR(9R + 2).$$

The condition $C_n - B_n = A_n^2$ becomes
$$cR(9R + 2) - bR = a^2 R^2.$$
Divide by $R \ne 0$ and rearrange:
$$9cR + 2c - b = a^2 R \quad\Longrightarrow\quad R\,(9c - a^2) = b - 2c.$$

The right side does not depend on $n$, while $R$ takes different values for different $n$. If $9c - a^2 \ne 0$, then $R = \dfrac{b - 2c}{9c - a^2}$ is a single value, so at most one $n$ works. Two values of $n$ therefore force
$$9c = a^2 \qquad\text{and}\qquad b = 2c,$$
and then the equation holds for every $n$.

$a^2 = 9c$ means $a$ is a multiple of $3$: $a = 3$ gives $c = 1$, $b = 2$ (sum $6$); $a = 6$ gives $c = 4$, $b = 8$ (sum $18$); $a = 9$ gives $c = 9$, $b = 18$, not a digit. The maximum is $6 + 8 + 4 = 18$.

Check with $n = 1$: $44 - 8 = 36 = 6^2$; with $n = 2$: $4444 - 88 = 4356 = 66^2$.

The answer is $\boxed{\textbf{(D)}\ 18}$.

Repdigits are multiples of repunits, and the $2n$-digit repunit factors through the $n$-digit one, so the whole equation lives in the single variable $R$. A polynomial identity in $R$ that must hold at two distinct values, when it is only linear in $R$, has to hold identically, which turns an infinite family of conditions into two digit equations. "At least two values of $n$" is the phrase that signals this coefficient-matching argument.

Reason from digits: $A_n^2 + B_n = C_n$. For $n = 1$ the equation is $a^2 + b = 11c$, and for $n = 2$ it is $121a^2 + 11b = 1111c$, i.e. $11a^2 + b = 101c$. Multiply the first by $11$ and subtract the second: $10b = 121c - 101c = 20c$, so $b = 2c$, and then $a^2 = 11c - b = 9c$, matching the general argument.

• Testing n = 1 alone (which yields many digit triples) or using C_n with n digits instead of 2n digits.
• Allowing $b = 18$ for $a = 9$, $c = 9$ (sum $36$, not a choice, but leading to a guess of (E)); digits must be single nonzero digits.