Difference between revisions of "2010 AMC 12B Problems/Problem 15"
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== Problem == | == Problem == | ||
− | For how many ordered triples <math>(x,y,z)</math> of nonnegative integers less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i | + | For how many ordered triples <math>(x,y,z)</math> of nonnegative integers less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i=\sqrt{-1}</math>? |
− | <math>(A) \ | + | <math>\textbf{(A)}\ 149 \qquad \textbf{(B)}\ 205 \qquad \textbf{(C)}\ 215 \qquad \textbf{(D)}\ 225 \qquad \textbf{(E)}\ 235</math> |
== Solution == | == Solution == |
Latest revision as of 17:29, 26 May 2023
Problem
For how many ordered triples of nonnegative integers less than
are there exactly two distinct elements in the set
, where
?
Solution
We have either ,
, or
.
only occurs when it is
.
has only one solution, namely,
.
has five solutions between 0 and 19,
, and
.
has nineteen integer solutions between zero and nineteen. So for
, we have
ordered triples.
For , again this only occurs at
.
has nineteen solutions,
has five solutions, and
has one solution, so again we have
ordered triples.
For , this occurs at
and
.
and
both have one solution while
has fifteen solutions.
and
both have one solution, namely,
and
, while
has twenty solutions (
only cycles as
). So we have
ordered triples.
In total we have ordered triples
Small Clarification
To more clearly see why the reasoning above is true, try converting the complex numbers into exponential form. That way, we can more easily raise the numbers to ,
and
respectively.
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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