Difference between revisions of "1997 USAMO Problems/Problem 6"

(added solution tag, USAMO box, and category.)
(Problem)
Line 2: Line 2:
 
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\lea_{i+j}\lea_i+a_j+1</math>
+
<math>a_i+a_j \l ea_{i+j} \le a_i+a_j+1</math>
  
for all <math>i, j\ge1</math> with <math>i+j\le1997</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\len\le1997</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>.
  
 
== Solution ==
 
== Solution ==

Revision as of 16:48, 13 June 2012

Problem

Suppose the sequence of nonnegative integers $a_1,a_2,...,a_{1997}$ satisfies

$a_i+a_j \l ea_{i+j} \le a_i+a_j+1$

for all $i, j \ge 1$ with $i+j \le 1997$. Show that there exists a real number $x$ such that $a_n=\lfloor{nx}\rfloor$ (the greatest integer $\lenx$ (Error compiling LaTeX. ! Undefined control sequence.)) for all $1 \le n \le 1997$.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See Also

1997 USAMO (ProblemsResources)
Preceded by
Problem 5
Followed by
Last Problem
1 2 3 4 5 6
All USAMO Problems and Solutions
Invalid username
Login to AoPS