if are positive real numbers satisfying then prove that
May 18, 2012
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December 8, 2011
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October 3, 2011
September 30, 2011
September 1, 2011
Problem 305 Van Khea
If are positive real numbers such that . Prove that
Proof
Using the well-known identity
Then the inequality above equivalent to
Let
The inequality transforms into
Which is clear true. Equality occurs for
August 23, 2011
Problem 304 Van Khea
If are positive real numbers such that then prove that
Solution
We have
Let then we get
Sitting yields
From inequality we have
and
then we get
Therefore the proof is completed. Equality occurs for
August 21, 2011
Problem 302 Van Khea
If are positive real numbers then prove that:
Solution
We have
Because then we have
Thus, it suffices to show that
Letting yields
From inequality we have
So we need to prove that
Now we will show that for are positive real numbers then we have
; and ;
without loss of generality, suppose that . Sitting and then:
which is obviously true. Equality occurs for
is true.
Therefore the proof is completed. Equality occurs for
Problem 301 Van Khea
If are positive real numbers such that . Prove that
Proof
Let then we get:
From inequality we have
Let
We have
Therefore we get
Because so from we get
Now we will prove that if be positive real numbers then
Let then we just need to prove that with then
From inequality we have
Therefore we get
Thus, it suffices to show that:
Which is just the third degree inequality
For
Therefore we get
From inequality we have
Therefore the proof is completed. Equality occurs for