Difference between revisions of "2009 AIME I Problems/Problem 1"
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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> | ||
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| + | == Video Solution by OmegaLearn == | ||
| + | https://youtu.be/1-iWPCWPsLw?t=195 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
==Video Solution== | ==Video Solution== | ||
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==Video Solution 2== | ==Video Solution 2== | ||
| − | https:// | + | https://www.youtube.com/watch?v=P00iOJdQiL4 |
~Shreyas S | ~Shreyas S | ||
Latest revision as of 02:32, 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. 
