1997 USAMO Problems/Problem 6

Revision as of 08:39, 1 July 2011 by Mcqueen (talk | contribs) (Created page with "== Problem == 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> for all <math>i, j\ge1</math>...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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. Unknown error_msg)

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. Unknown error_msg)) for all $1\len\le1997$ (Error compiling LaTeX. Unknown error_msg).

Solution