Difference between revisions of "2002 AIME II Problems/Problem 3"
(→Solution) |
Flyhawkeye (talk | contribs) (→Solution: Added LaTeX and clarified solution) |
||
Line 5: | Line 5: | ||
<math>abc=6^6</math>. Since they form an increasing geometric sequence, <math>b</math> is the [[geometric mean]] of the [[product]] <math>abc</math>. <math>b=\sqrt[3]{abc}=6^2=36</math>. | <math>abc=6^6</math>. Since they form an increasing geometric sequence, <math>b</math> is the [[geometric mean]] of the [[product]] <math>abc</math>. <math>b=\sqrt[3]{abc}=6^2=36</math>. | ||
− | Since <math>b-a</math> is the square of an integer, we can find a few values of <math>a</math> that work: 11, 20, 27, 32, and 35. | + | Since <math>b-a</math> is the square of an integer, we can find a few values of <math>a</math> that work: <math>11, 20, 27, 32,</math> and <math>35</math>. Out of these, the only value of <math>a</math> that works is <math>a=27</math>, from which we can deduce that <math>c=\dfrac{36}{27}\cdot 36=\dfrac{4}{3}\cdot 36=48</math>. |
− | <math>a+b+c=27+36+48=\boxed{111}</math> | + | Thus, <math>a+b+c=27+36+48=\boxed{111}</math> |
== See also == | == See also == | ||
{{AIME box|year=2002|n=II|num-b=2|num-a=4}} | {{AIME box|year=2002|n=II|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 11:03, 24 July 2018
Problem
It is given that where
and
are positive integers that form an increasing geometric sequence and
is the square of an integer. Find
Solution
. Since they form an increasing geometric sequence,
is the geometric mean of the product
.
.
Since is the square of an integer, we can find a few values of
that work:
and
. Out of these, the only value of
that works is
, from which we can deduce that
.
Thus,
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.