[/quote]
,
?
be three positive reals such that 
.![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
real numbers 
,
and transform the sequence into 
.
where 
are lengths of parallel sides and 
are other two sides.
be a cyclic quadrilateral. Let 
,
,
be the feet of the perpendiculars from 
to the lines 
,
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
be reals such that 
. Prove that
.
be the feet of the altitudes from 
to 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
and 
Prove that 
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
.
if and only if 
.
and 
.
be reals such that 
Prove that
Prove that
such that 
.
Y by teomihai
Suppose you have an 
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).This post has been edited 2 times. Last edited by Anto0110, Apr 5, 2025, 1:58 PM
Reason: 9
Reason: 9
