[/quote]
,
.![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](http://latex.artofproblemsolving.com/9/0/3/903220c471bdd3028dda2fcb13874a4bcf752872.png)
without using the Rearrangement Inequality or Chebyshev's Inequality.
which satisfy the following set of equations
inscribed in 
with 
being the midpoint of 
.
of 
.
be the projection of 
onto 
.
that is not on 
intersects 
at 
.
and 
intersect at 
.
is an isosceles triangle.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
such that 
divides 
for any positive integers 
and 
.
of real numbers such that 
for all positive integers 
.
denotes the greatest integer less than or equal to 
.
distinct real numbers there exist two pairs 
and 
with 
or 
,![\[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]](http://latex.artofproblemsolving.com/3/b/0/3b03250ffbd03fd657fa00bc2b63a2c17ee75061.png)
be an arbitrary diameter of its circumscribed circle, and the point 
lies on the smaller arc 
,
is on the smaller arc 
.
are feet of perpendiculars dropped from point 
to line 
lies on a fixed circle.
,
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This post has been edited 19 times. Last edited by jwelsh, Mar 21, 2020, 5:28 PM
Reason: update
Reason: update
