[/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 
,
,
are selected on sides 
,
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 
,
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
A geometry program on the computer allows the following operations to be performed:
Mark points on segments, on lines, or outside them.
Draw the line that joins two points.
Find the point of intersection of two lines.
Given a point 
and a line 
, trace the symmetric of with respect to .
Given an
triangle, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.
Mark points on segments, on lines, or outside them. Draw the line that joins two points. Find the point of intersection of two lines. Given a point 
and a line 
, trace the symmetric of with respect to .Given an
triangle, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.This post has been edited 1 time. Last edited by parmenides51, Jun 16, 2020, 4:37 PM
as the intersection point of 
and 
,
and 
are the symmetric points of 
,
the symmetric points of 
implies that 
.
is the perpendicular bisector of 
and 
,
