Problem 1.
If a, b, c are non-negative real numbers such that , then prove that the product abc cannot exceed 1.
Solution I:
Given that ,
,
, so certainly
,
,
, and
.
Now, and hence,
, hence we get:
.ย Clearly, the presence ofย
and
reminds us of the AM-GM inequality.
Here it is .
So, .
Also, we can say: . Now, let
.
So,
that is, , or
, that is,
.ย So, this is a beautiful application of arithmetic mean-geometric mean inequality twice. ๐ ๐
Problem 2:
If a, b, c are three rational numbers, then prove that : is always the square of a rational number.
Solution 2:
Let ,
,
. It can be very easily shown that
, or
. So, the given expression
is a perfect square !!!ย BINGO! ๐ ๐ ๐
Nalin Pithwa.