Difference between revisions of "2023 AIME II Problems/Problem 2"
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== Solution == | == Solution == | ||
| + | Assuming that such palindrome is greater than <math>777_8 = 511,</math> we conclude that the palindrome has four digits when written in base eight. Let such palindrome be <cmath>(\underline{ABBA})_8 = 512A + 64B + 8B + A = 513A + 72B.</cmath> | ||
| + | It is clear that <math>A=1,</math> so we repeatedly add <math>72</math> to <math>513</math> until we get palindromes: | ||
| + | \begin{align*} | ||
| + | 513+72\cdot0 &= 513, \\ | ||
| + | 513+72\cdot1 &= \boxed{585}, \\ | ||
| + | 513+72\cdot2 &= 657, \\ | ||
| + | 513+72\cdot3 &= 729, \\ | ||
| + | 513+72\cdot4 &= 801, \\ | ||
| + | 513+72\cdot5 &= 873, \\ | ||
| + | 513+72\cdot6 &= 945, \\ | ||
| + | 513+72\cdot7 &= 1017. \\ | ||
| + | \end{align*} | ||
== See also == | == See also == | ||
{{AIME box|year=2023|num-b=1|num-a=3|n=II}} | {{AIME box|year=2023|num-b=1|num-a=3|n=II}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 16:31, 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
Assuming that such palindrome is greater than 
we conclude that the palindrome has four digits when written in base eight. Let such palindrome be ![\[(\underline{ABBA})_8 = 512A + 64B + 8B + A = 513A + 72B.\]](http://latex.artofproblemsolving.com/8/9/0/89097713b51c8d9e9063d1dca16394cd50efe1ef.png)
It is clear that 
so we repeatedly add 
to 
until we get palindromes:
\begin{align*}
513+72\cdot0 &= 513, \\
513+72\cdot1 &= \boxed{585}, \\
513+72\cdot2 &= 657, \\
513+72\cdot3 &= 729, \\
513+72\cdot4 &= 801, \\
513+72\cdot5 &= 873, \\
513+72\cdot6 &= 945, \\
513+72\cdot7 &= 1017. \\
\end{align*}
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. 
