1989 AIME Problems/Problem 14

Problem

Given a positive integer $n^{}_{}$ , it can be shown that every complex number of the form $r+si^{}_{}$ , where $r^{}_{}$ and $s^{}_{}$ are integers, can be uniquely expressed in the base $-n+i^{}_{}$ using the integers $1,2^{}_{},\ldots,n^2$ as digits. That is, the equation

$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$

is true for a unique choice of non-negative integer $m^{}_{}$ and digits $a_0,a_1^{},\ldots,a_m$ chosen from the set $\{0^{}_{},1,2,\ldots,n^2\}$ , with $a_m\ne 0^{}^{}$ (Error compiling LaTeX. Unknown error_msg). We write

$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$

to denote the base $-n+i^{}_{}$ expansion of $r+si^{}_{}$ . There are only finitely many integers $k+0i^{}_{}$ that have four-digit expansions

$k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.$

Find the sum of all such $k^{}_{}$ .

Solution

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1989 AIME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1989 AIME Problems/Problem 14

Problem

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