1994 IMO Problems/Problem 1
Let and
be two positive integers. Let
,
,
,
be
different numbers from the set
such that for any two indices
and
with
and
, there exists an index
such that
. Show that
.
Solution
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
See Also
1994 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |