1989 AIME Problems/Problem 14

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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 $a_m\ne 0$ . 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

First, we find the first three powers of $-3+i$ :

$(-3+i)^1=-3+i ; (-3+i)^2=8-6i ; (-3+i)^3=-18+26i$

So we need to solve the diophantine equation $a_1-6a_2+26a_3=0 \Longrightarrow a_1-6a_2=-26a_3$ .

The minimum the left hand side can go is -54, so $a_3\leq 2$ , so we try cases:

Case 1: $a_3=2$

The only solution to that is $(a_1, a_2, a_3)=(2,9,2)$ .

Case 2: $a_3=1$

The only solution to that is $(a_1, a_2, a_3)=(4,5,1)$ .

Case 3: $a_3=0$

$a_3$ cannot be 0, or else we do not have a four digit number.

So we have the four digit integers $(292a_0)_{-3+i}$ and $(154a_0)_{-3+i}$ , and we need to find the sum of all integers $k$ that can be expressed by one of those.

$(292a_0)_{-3+i}$ :

We plug the first three digits into base 10 to get $30+a_0$ . The sum of the integers $k$ in that form is $345$ .

$(154a_0)_{-3+i}$ :

We plug the first three digits into base 10 to get $10+a_0$ . The sum of the integers $k$ in that form is $145$ . The answer is $345+145=\boxed{490}$ .

1989 AIME (Problems • 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

From AoPSWiki

Problem

Solution

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