Difference between revisions of "2009 AIME I Problems/Problem 1"
(→Solution 2) |
|||
Line 11: | Line 11: | ||
=== Solution 2 === | === Solution 2 === | ||
− | + | Consider the three-digit number <math>\overline{abc}</math>. If its digits form a geometric sequence, we must have that <math>{a \over b} = {b \over c}</math>, that is, <math>b^2 = ac</math>. | |
− | + | The minimum and maximum geometric numbers occur when <math>a</math> is minized and maximized, respectively. The minimum occurs when <math>a = 1</math>; letting <math>b = 2</math> and <math>c = 4</math> achieves this, so the smallest possible geometric number is 124. | |
+ | For the maximum, we have that <math>b^2 = 9c</math>; <math>b</math> is maximized when <math>9c</math> is the greatest possible perfect square; this happens when <math>c = 4</math>, yielding <math>b = 6</math>. Thus, the largest possible geometric number is 964. | ||
+ | |||
+ | Our answer is thus <math>964 - 124 = \boxed{840}</math>. | ||
== See also == | == See also == |
Revision as of 16:00, 21 March 2009
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
Solution 1
Assume that the largest geometric number starts with a nine. 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 sequence, we must have that , that is, .
The minimum and maximum geometric numbers occur when is minized 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 .
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 |