2007 IMO Problems/Problem 1
Problem 1
Real numbers are given. For each
(
) define
and let
.
(a) Prove that, for any real numbers ,
(b) Show that there are real numbers such that equality holds in (*)
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 a 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 {x_i} 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
,
This is written by Mo Lam--- who is a horrible proof writer, so please fix the proof for me. Thank you. O, also the formatting.