Difference between revisions of "1989 AIME Problems/Problem 14"
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== Problem == | == Problem == | ||
| + | Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation | ||
| + | <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> | ||
| + | is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write | ||
| + | <center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center> | ||
| + | to denote the base <math>-n+i^{}_{}</math> expansion of <math>r+si^{}_{}</math>. There are only finitely many integers <math>k+0i^{}_{}</math> that have four-digit expansions | ||
| + | <center><math>k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.</math></center> | ||
| + | Find the sum of all such <math>k^{}_{}</math>. | ||
== Solution == | == Solution == | ||
| + | {{solution}} | ||
== See also == | == See also == | ||
| + | * [[1989 AIME Problems/Problem 15|Next Problem]] | ||
| + | * [[1989 AIME Problems/Problem 13|Previous Problem]] | ||
* [[1989 AIME Problems]] | * [[1989 AIME Problems]] | ||
Revision as of 22:24, 24 February 2007
Problem
Given a positive integer 
, it can be shown that every complex number of the form 
, where 
and 
are integers, can be uniquely expressed in the base 
using the integers 
as digits. That is, the equation
is true for a unique choice of non-negative integer 
and digits 
chosen from the set 
, with 
. We write
to denote the base expansion of . There are only finitely many integers 
that have four-digit expansions
Find the sum of all such 
.
Solution
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