1994 IMO Problems/Problem 1
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Let satisfy the given conditions. We will prove that for all
WLOG, let . Assume that for some
This implies, for each
because
For each of these values of i, we must have such that
is a member of the sequence for each
. Because
.
Combining all of our conditions we have that each of
must be distinct integers such that
However, there are distinct
, but only
integers satisfying the above inequality, so we have a contradiction. Our assumption that
was false, so
for all
such that
Summing these inequalities together for
gives
which rearranges to