Difference between revisions of "1997 USAMO Problems/Problem 6"
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Suppose the sequence of nonnegative integers <math>a_1,a_2,...,a_{1997}</math> satisfies | Suppose the sequence of nonnegative integers <math>a_1,a_2,...,a_{1997}</math> satisfies | ||
| − | <math>a_i+a_j \ | + | <math>a_i+a_j \le a_{i+j} \le a_i+a_j+1</math> |
for all <math>i, j \ge 1</math> with <math>i+j \le 1997</math>. Show that there exists a real number <math>x</math> such that <math>a_n=\lfloor{nx}\rfloor</math> (the greatest integer <math>\lenx</math>) for all <math>1 \le n \le 1997</math>. | for all <math>i, j \ge 1</math> with <math>i+j \le 1997</math>. Show that there exists a real number <math>x</math> such that <math>a_n=\lfloor{nx}\rfloor</math> (the greatest integer <math>\lenx</math>) for all <math>1 \le n \le 1997</math>. | ||
Revision as of 16:49, 13 June 2012
Problem
Suppose the sequence of nonnegative integers 
satisfies
for all 
with 
. Show that there exists a real number 
such that 
(the greatest integer $\lenx$ (Error compiling LaTeX. Unknown error_msg)) for all 
.
Solution
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See Also
| 1997 USAMO (Problems • Resources) | ||
| Preceded by Problem 5 |
Followed by Last Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
