Difference between revisions of "1989 AIME Problems/Problem 14"
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== Problem == | == Problem == | ||
− | Given a positive [[integer]] <math>n | + | 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> | <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 | + | 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> | <center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center> | ||
− | to denote the base <math>-n+i | + | 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 | + | <center><math>k=(a_3a_2a_1a_0)_{-3+i}~~</math><math>~~a_3\ne 0.</math></center> |
− | Find the sum of all such <math>k | + | Find the sum of all such <math>k</math>. |
== Solution == | == Solution == |
Revision as of 22:14, 17 July 2008
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
![$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$](http://latex.artofproblemsolving.com/f/6/4/f6491a5fef34ba26ef6657aae299c94700b40a0e.png)
is true for a unique choice of non-negative integer and digits
chosen from the set
, with
. We write
![$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$](http://latex.artofproblemsolving.com/5/6/a/56aa9ca7c641491fbd454fff558b7a8fb8918ced.png)
to denote the base expansion of
. There are only finitely many integers
that have four-digit expansions
![$k=(a_3a_2a_1a_0)_{-3+i}~~$](http://latex.artofproblemsolving.com/f/8/1/f81d425eb93665b6c8598c7cf275730089c4df4d.png)
![$~~a_3\ne 0.$](http://latex.artofproblemsolving.com/0/6/3/063a24c5b1954678bd93bbec326e554e671a4acf.png)
Find the sum of all such .
Solution
First, we find the first three powers of :
So we need to solve the diophantine equation .
The minimum the left hand side can go is -54, so , so we try cases:
- Case 1:
- The only solution to that is
.
- Case 2:
- The only solution to that is
.
- Case 3:
cannot be 0, or else we do not have a four digit number.
So we have the four digit integers and
, and we need to find the sum of all integers
that can be expressed by one of those.
:
We plug the first three digits into base 10 to get . The sum of the integers
in that form is
.
:
We plug the first three digits into base 10 to get . The sum of the integers
in that form is
. The answer is
.
See also
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 | ||
All AIME Problems and Solutions |