be an acute scalene triangle inscribed in a circle 
.
intersects 
at 
and 
.
is the midpoint of 
.
be the altitude in 
,
intersects 
.
be the midpoint of 
be the reflection of 
with respect to 
and 
are similar.
Y by adityaguharoy, Adventure10, Mango247
Let 
be positive real numbers . Find the minimum value of 
Let
and 
Find the maximum value of 
be positive real numbers . Find the minimum value of 
Let
and 
Find the maximum value of 
.
.
.
Hence shifting any 
to 
is non-increasing, so we may assume 
.
may be done with standard calculus tricks.
Thus we just minimise the function 
over 
which can be done with standard calculus tricks.
implies that
where the last inequality holds because 
for all 
,
.
is 
.
Let 
Proof:
,
But 
when 
,
is 
.
is fixed positive real. Let 
in terms of 
,
But 
when 
,
is 
.
Let 
![\[a^3+ b^3-kab + \frac{k^3}{27} = \frac{(3b+3a+k)}{27} \cdot \frac{(6a-3b-k)^2+3(3b-k)^2}4.\]](http://latex.artofproblemsolving.com/2/8/5/2859187e2d5806549f3b7008886a518160c84b52.png)
So![\[a^3+ b^3-kab \ge -\frac{k^3}{27}.\]](http://latex.artofproblemsolving.com/e/8/f/e8fa110c024aca972b043a4d2e27e359eff97039.png)
![$$a^3+ b^3-kab\geq 3\sqrt[3]{a^3\cdot b^3\cdot\frac{k^3}{27}}-kab-\frac{k^3}{27}=-\frac{k^3}{27}.$$](http://latex.artofproblemsolving.com/c/5/f/c5ff0bed94583e9cba00e7118483debb07a75332.png)
The equality occurs when 
.
,
,
be positive reals such that 
Show that
and 
,
thereby 
.
We are done. 
be positive reals such that 
Prove that 
,
be positive reals such that 
Show that
,
which yields 
。
As stated. 
and 
we obtain the quoted result.
be positive reals such that 
Then 
Prove that 
.
.
and the inequality is equivalent to 
which is obvious.
be positive reals such that 
Find the maximum value of 
.
where 
.
.
.
.
It's clear that for positive 
is increasing, so maximum of 
.
Find the minimum and maximum value of 
and 
Prove that
Old.
Prove that
