AIME 1989 · 第 14 题

AIME 1989 — Problem 14

专题: Contest Math
难度: L4
来源: AIME

题目详情

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 $0,1,2,\ldots,n^2$ 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 $k$ ,

解析

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 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 $1\leq a_3 \leq 2$ since $a_3$ can't equal 0, 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 $(a_1, a_2, a_3)=(4,5,1)$ .

So we have 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 $345+145=\boxed{490}$ . ~minor edit by Yiyj1