.![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](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.
,
Y by
Let the incircle of 
touch sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.
(Kritesh Dhakal, Nepal)
touch sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.(Kritesh Dhakal, Nepal)
This post has been edited 2 times. Last edited by jenishmalla, Mar 15, 2025, 3:00 PM
Reason: formatting
Reason: formatting
and take a homography preserving the circle and taking 
to the centre of the circle, from where it's obvious.
.
as the incenter of 
and let 
.
Since 
,
lies in a circle with diameter 
.
is a cyclic pentagon.
,
.
,
and 
is a harmonic quadrilateral. Brianchon's Theorem on hexagon 
,
and 
concur on line 
.
,
be the center of spiral similarity that sends 
that 
as 
meet the incircle again at 
to be the unit circle.
.
are collinear 
.
are collinear 
.
.
,
and we're done.
say 
(and 
)
lies on 
by La Hire. Now 
,
,
.
finishes.
![\[
P = EF \cap AD, \quad Q = EF \cap AD'.
\]](http://latex.artofproblemsolving.com/4/c/4/4c43ce3efdb65c792f4ba9800e4cc4b4742eef74.png)
![\[
(A, Q; D', X) = (A, P; Y, D) = -1.
\]](http://latex.artofproblemsolving.com/1/4/2/142bb90c50b44b6dfe0c36d97d43bdfe10f3b5cc.png)
and 
are harmonic divisions.
and 
and if 
