Difference between revisions of "2024 USAJMO Problems/Problem 3"
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== Problem == | == Problem == | ||
Let <math>a(n)</math> be the sequence defined by <math>a(1)=2</math> and <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>. | Let <math>a(n)</math> be the sequence defined by <math>a(1)=2</math> and <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>. | ||
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| + | == Solution 1 == | ||
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| + | ==See Also== | ||
| + | {{USAJMO newbox|year=2024|num-b=2|num-a=4}} | ||
| + | {{MAA Notice}} | ||
Revision as of 21:44, 19 March 2024
Contents
Problem
Let 
be the sequence defined by 
and 
for each integer 
. Suppose that 
is prime and 
is a positive integer. Prove that some term of the sequence is divisible by 
.
Solution 1
See Also
| 2024 USAJMO (Problems • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
