Difference between revisions of "2009 AIME I Problems/Problem 1"
Pi is 3.14 (talk | contribs) (→Video Solution) |
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==Solution 3== | ==Solution 3== | ||
The smallest geometric number is <math>124</math> because <math>123</math> and any number containing a zero does not work. <math>964</math> is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives <math>\boxed{840}.</math> | The smallest geometric number is <math>124</math> because <math>123</math> and any number containing a zero does not work. <math>964</math> is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives <math>\boxed{840}.</math> | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
+ | https://youtu.be/1-iWPCWPsLw?t=195 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==Video Solution== | ==Video Solution== |
Latest revision as of 03:32, 16 January 2023
Contents
Problem
Call a -digit number geometric if it has
distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
Solution 1
Assume that the largest geometric number starts with a . We know that the common ratio must be a rational of the form
for some integer
, because a whole number should be attained for the 3rd term as well. When
, the number is
. When
, the number is
. When
, we get
, but the integers must be distinct. By the same logic, the smallest geometric number is
. The largest geometric number is
and the smallest is
. Thus the difference is
.
Solution 2
Consider the three-digit number . If its digits form a geometric progression, we must have that
, that is,
.
The minimum and maximum geometric numbers occur when is minimized and maximized, respectively. The minimum occurs when
; letting
and
achieves this, so the smallest possible geometric number is 124.
For the maximum, we have that ;
is maximized when
is the greatest possible perfect square; this happens when
, yielding
. Thus, the largest possible geometric number is 964.
Our answer is thus .
Solution 3
The smallest geometric number is because
and any number containing a zero does not work.
is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives
Video Solution by OmegaLearn
https://youtu.be/1-iWPCWPsLw?t=195
~ pi_is_3.14
Video Solution
~IceMatrix
Video Solution 2
https://www.youtube.com/watch?v=P00iOJdQiL4
~Shreyas S
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
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.