Difference between revisions of "2007 USAMO Problems/Problem 1"
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== Problem == | == Problem == | ||
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| + | Let <math>n</math> be a positive integer. Define a sequence by setting <math>a_1 = n</math> and, for each <math>k>1</math>, letting <math>a_k</math> be the unique integer in the range <math>0 \le a_k \le k-1</math> for which <math>a_1 + a_2 + \cdots + a_k</math> is divisible by <math>k</math>. For instance, when <math>n=9</math> the obtained sequence is <math>9, 1, 2, 0, 3, 3, 3, \ldots</math>. Prove that for any <math>n</math> the sequence <math>a_1, a_2, a_3, \ldots</math> eventually becomes constant. | ||
== Solution == | == Solution == | ||
{{USAMO newbox|year=2007|before=First question|num-a=2}} | {{USAMO newbox|year=2007|before=First question|num-a=2}} | ||
Revision as of 16:56, 25 April 2007
Problem
Let 
be a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.
Solution
| 2007 USAMO (Problems • Resources) | ||
| Preceded by First question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
