Difference between revisions of "2022 AIME I Problems/Problem 2"
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Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits. | Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits. | ||
| − | == Solution == | + | == Solution 1 == |
We are given that <cmath>100a + 10b + c = 81b + 9c + a,</cmath> which rearranges to <cmath>99a = 71b + 8c.</cmath> | We are given that <cmath>100a + 10b + c = 81b + 9c + a,</cmath> which rearranges to <cmath>99a = 71b + 8c.</cmath> | ||
Revision as of 00:04, 18 February 2022
Contents
Problem
Find the three-digit positive integer 
whose representation in base nine is 
where 
and 
are (not necessarily distinct) digits.
Solution 1
We are given that ![\[100a + 10b + c = 81b + 9c + a,\]](http://latex.artofproblemsolving.com/9/8/4/98419c4a18d61bbd8b2c652976c8427a082c8167.png)
which rearranges to ![\[99a = 71b + 8c.\]](http://latex.artofproblemsolving.com/0/2/a/02a0693b866a5a516e33b7d14ea36547a9b2e122.png)
Taking both sides modulo 
we have
The only solution occurs at 
from which 
Therefore, the requested three-digit positive integer is 
~MRENTHUSIASM
Solution 2 (Simple)
As in the previous solution, we get 
. 99 and 71 are big numbers comparatively to 8, so we hypothesize that 
and 
are equal and 
fills the gap between them. The difference between 99 and 71 is 28, which is a multiple of 4, so if we multiply this by 2, it will be a multiple of 8 and thus the gap can be filled. Therefore, a viable solution is 
and the answer is 
~KingRavi
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=z5Y4bT5rL-s
Video Solution
https://www.youtube.com/watch?v=CwSkAHR3AcM
~Steven Chen (www.professorchenedu.com)
See Also
| 2022 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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. 
