AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
be a triangle with incenter 
, and circumcircle 
. Let the circle with diameter 
intersects at 
and 
. Let 
be the midpoint arc 
and let 
intersect again at a point 
. Let the circumcircle of 
intersect the circumcircle of 
at a point 
. Let the tangent at to intersect 
at a point 
. Prove that 
and 
intersect on .
satisfy,
Where {
} is a permutation of . Prove that 
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 fields of size 
each field being either yellow or blue. No two countries have the same flag. We say that a set of flags is diverse if these flags can be arranged into an 
square so that all fields on its main diagonal will have the same color. Determine the smallest positive integer such that among any distinct flags, there exist flags forming a diverse set.
, 
. Let the incenter, 
the circumcircle of triangle , and the foot of perpendicular from to . intersects at point 
, and the line which passes and perpendicular to 
intersects at point 
. Now let 
the foot of perpendicular from to .
.
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 
, one can reach any other black cell 
through a sequence 
of black cells where each consecutive pair 
are adjacent (sharing a common side) for every 
. Let 
denote the maximum possible cost over all initial colorings with exactly black cells. Determine a constant 
such that![\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]](http://latex.artofproblemsolving.com/0/a/0/0a06d4ff52479c8cb785c6450cea8748bf7769b6.png)
for any 
.
. Let 
be an interior point such that 
, 
and 
. Find 
.
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 , isogonal conjugate 
lies on the McCay cubic. Let 
be the pedal triangle of wrt and let 
. Let 
be the second intersection of 
with 
and let be the Poncelet point of 
. Then the lines 
are concurrent.
and let 
be the second intersection of with 
be the pedal triangle of 
wrt . Then 
.
be the circumcevian triangle of wrt . Then 
and 
are homothetic.
. On the other hand, from the Lemma 1, we see that 
. Therefore from the Lemma 2, we see that 
and 
, respectively, are collinear and is the insimilicenter of 
and , so 
. of is tangent to at . Hence by applying Monge-D'Alembert's theorem to 
, we see that 
are collinear.
, 
. Prove that
![$[BH]$](http://latex.artofproblemsolving.com/a/9/f/a9fd89d48264ff1c3f9b77f62ef66f916978d99d.png)
dropped onto the hypotenuse of a triangle intersects the bisectors ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
and ![$[CE]$](http://latex.artofproblemsolving.com/e/a/8/ea8eadb930f32cceb220dee99277c24b202fb80c.png)
at and respectively. Prove that the line passing through the midpoints of the segments ![$[QD]$](http://latex.artofproblemsolving.com/2/b/2/2b24b6b34a55e29ef31a8cc9a72f116d3896a69f.png)
and ![$[PE]$](http://latex.artofproblemsolving.com/7/9/2/792976015f6d59bd3b89b6d2d53901be85cecd92.png)
is parallel to the line 
.
dropped onto the hypotenuse of a triangle intersects the bisectors and at and respectively. Prove that the line passing through the midpoints of the segments and is parallel to the line .
are isosceles, hence if 
are the midpoints of , respectively , we have 
, 
, i.e. 
is cyclic, being the incenter. From 
. From we get 
, or 
.
and 
and 
.After that it is just simple calculations and not much thinking.
?
Does EV theorem work?
