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

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== Problem ==
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Suppose the sequence of nonnegative integers <math>a_1,a_2,...,a_{1997}</math> satisfies
  
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<math>a_i+a_j\lea_{i+j}\lea_i+a_j+1</math>
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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>.
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== Solution ==

Revision as of 14:09, 5 July 2011

Problem

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

$a_i+a_j\lea_{i+j}\lea_i+a_j+1$ (Error compiling LaTeX. ! Undefined control sequence.)

for all $i, j\ge1$ with $i+j\le1997$. 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 $1\len\le1997$ (Error compiling LaTeX. ! Undefined control sequence.).

Solution

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