Difference between revisions of "2023 AIME II Problems/Problem 2"
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| − | Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than <math>1000</math> that is a palindrome both when written in base ten and when written in base eight, such as <math>292 = 444_{eight}</math>. | + | == Problem == |
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| + | Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than <math>1000</math> that is a palindrome both when written in base ten and when written in base eight, such as <math>292 = 444_{\text{eight}}.</math> | ||
| + | |||
| + | == Solution == | ||
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| + | If the palindrome written in base eight has three digits, then we have <cmath>(\underline{ABA})_8 = 64A+8B+A = 65A+8B \leq 511.</cmath> | ||
| + | If the palindrome written in base eight has four digits, then we have <cmath>(\underline{ABBA})_8 = 512A+64B+8B+A = 513A+72B \geq 513.</cmath> | ||
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| + | == See also == | ||
| + | {{AIME box|year=2023|num-b=1|num-a=3|n=II}} | ||
| + | {{MAA Notice}} | ||
Revision as of 15:10, 16 February 2023
Problem
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than 
that is a palindrome both when written in base ten and when written in base eight, such as 
Solution
If the palindrome written in base eight has three digits, then we have ![\[(\underline{ABA})_8 = 64A+8B+A = 65A+8B \leq 511.\]](http://latex.artofproblemsolving.com/1/5/2/15292462e7675bcc36e3a8b1a24d9652a3ae5714.png)
If the palindrome written in base eight has four digits, then we have ![\[(\underline{ABBA})_8 = 512A+64B+8B+A = 513A+72B \geq 513.\]](http://latex.artofproblemsolving.com/3/b/8/3b812fe4cf34b26badae88ee659f522df768b4d6.png)
See also
| 2023 AIME II (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. 
