2007 IMO Shortlist Problems/A1
Problem
(New Zealand)
You are given a sequence of numbers. For each
(
) define
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and let

(a) Prove that for arbitrary real numbers ,

(b) Show that there exists a sequence of real numbers such that we have equality in (a).
Solution
Since , all
can be expressed as
, where
.
Thus,
can be expressed as
for some
and
,
Lemma)
Assume for contradiction that
, then for all
,
Then,
is a non-decreasing function, which means,
, and
, which means,
.
Then,
and contradiction.
a) Case 1)
If
,
is the maximum of a set of non-negative number, which must be at least
.
Case 2)
(We can ignore
because of lemma)
Using the fact that
can be expressed as
for some
and
,
.
Assume for contradiction that
.
Then,
,
.
, and
Thus,
and
.
Subtracting the two inequality, we will obtain:
--- contradiction (
).
Thus,
(b) A set of where the equality in (*) holds is:
Since
is a non-decreasing function,
is non-decreasing.
:
Let
,
.
Thus,
(
because
is the max of a set including
)
Since
and
,
.\
(Taken from https://artofproblemsolving.com/wiki/index.php/2007_IMO_Problems/Problem_1)