ISL 2005 A3
by Wolstenholme, Sep 6, 2014, 2:25 AM
Four real numbers 
, 
, 
, 
satisfy 
and 
. Prove that there exists a permutation 
of 
such that 
.
Proof:
Assume WLOG that
. Note that 
. By the rearrangement inequality, we have that 
and so 
. Letting 
we have that 
which becomes 
. This means that 
. But since this means that 
and so we are done.
Now I want to discuss motivation. After some playing around we see the only equality case is
. Now, looking at this, we want the extreme values of the sum of the two biggest or the two smallest to be greater than 
and less than 
respectively, so letting we want to get exactly the equation 
. Now we want an 
in there somewhere, and rearranging it turns out we want 
But this is the same as , and when looking at a sum like this, the rearrangement inequality should come to mind immediately.
, 
, 
, 
satisfy 
and 
. Prove that there exists a permutation 
of 
such that 
.Proof:
Assume WLOG that
. Note that 
. By the rearrangement inequality, we have that 
and so 
. Letting 
we have that 
which becomes 
. This means that 
. But since this means that 
and so we are done.Now I want to discuss motivation. After some playing around we see the only equality case is
. Now, looking at this, we want the extreme values of the sum of the two biggest or the two smallest to be greater than 
and less than 
respectively, so letting we want to get exactly the equation 
. Now we want an 
in there somewhere, and rearranging it turns out we want 
But this is the same as , and when looking at a sum like this, the rearrangement inequality should come to mind immediately.
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