1997 USAMO Problems/Problem 6

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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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