[/quote]
,
be real numbers . Prove that
![$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$](http://latex.artofproblemsolving.com/9/9/9/9997397eb38cd4bceec2ff8c85951c78e2797d0f.png)
be triangle with circumcenter 
and orthocenter 
.
intersect 
respectively. Lines through 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
be a triangle with incenter 
,
intersects 
and 
be the midpoint arc 
and let 
intersect 
.
intersect the circumcircle of 
at a point 
.
at a point 
.
and 
intersect on 
satisfy,
Where 
}
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
countries 
Each country has a flag 
units wide and one unit high composed of 
each field being either yellow or blue. No two countries have the same flag. We say that a set of 
square so that all 
.
the circumcircle of triangle 
,
intersects 
.
the foot of perpendicular from 
.
?
cells of an 
square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell 
,
through a sequence 
of black cells where each consecutive pair 
are adjacent (sharing a common side) for every 
.
denote the maximum possible cost over all initial colorings with exactly 
such that![\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]](http://latex.artofproblemsolving.com/0/a/0/0a06d4ff52479c8cb785c6450cea8748bf7769b6.png)
for any 
.
Does EV theorem work?
,
and 
.
.
be real numbers satisfying 
. Find the maximum value of![\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]](http://latex.artofproblemsolving.com/7/b/2/7b2ec0a87ca05d3cb29da888a4dfa73e10f238f8.png)
with orthocenter 
lies on the McCay cubic. Let 
be the pedal triangle of 
.
be the second intersection of 
with 
and let 
.
are concurrent.
and let 
be the second intersection of 
be the pedal triangle of 
.
be the circumcevian triangle of 
and 
are homothetic.
.
.
and 
,
and 
.
,
are collinear.
, 
.
Y by Mango247
The quadrilateral 
is inscribed in a circle. At point 
, tangent 
is drawn to this circle. Circle 
passes through points 
and 
and touches line at point 
. Line 
intersects 
at point 
. Find the ratio 
if it is known that 
is tangent to circle .
is inscribed in a circle. At point 
, tangent 
is drawn to this circle. Circle 
passes through points 
and 
and touches line at point 
. Line 
intersects 
at point 
. Find the ratio 
if it is known that 
is tangent to circle .
