[/quote]
,
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
and circumcenter 
.
,
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and incenter 
and let 
,
,
such that 
,
,
.
intersects 
other than 
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its circumradius. Let 
across the line 
,
across the line 
.
.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
.
is a prime number
is inscribed in circle 
.
its 
.
meets 
at 
and 
meets 
a second time at 
meets 
a second time at 
.
passes through the antipodal to 
intersects the lines 
,
at points 
,
,
of the segments 
,
,
,
.
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
Y by Adventure10, Mango247
Points 
and 
are chosen on the bisector of angle 
of a triangle 
in such a way that 
. Prove that
(a)
,
(b) the circle that passes through and and is tangent to 
is also tangent to the circle circumscribed around triangle .
and 
are chosen on the bisector of angle 
of a triangle 
in such a way that 
. Prove that(a)
,(b) the circle that passes through
and and is tangent to 
is also tangent to the circle circumscribed around triangle .
,
