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.
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
such that 
and there exists an integer 
such that 
is a multiple of 
.
is chosen on the side 
of a rectangle 
. From the vertex 
, the perpendicular 
is dropped to the segment 
. The segments and divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to are equal then the third circle is also equal to them.
be a non-constant function from the set of positive integers into the set of positive integer, such that 
divides 
for all distinct positive integers 
, 
. Prove that there exist infinitely many primes 
such that divides 
for some positive integer 
.
of degree 
has exactly 
real roots, counted with multiplicity. We know that the coefficient of 
is 
, all the roots are less than or equal to , and 
. What values can 
take?
lies inside triangle 
such that 
and 
. Point 
is the midpoint of segment 
. Point 
lies on segment 
with 
. Prove that 
.
, and 
of a convex hexagon 
intersect in a common point.
,
is the area of the triangle 
.
?
Does EV theorem work?
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)
and 
.![\[ab+bc+ac\leqslant 27\]](http://latex.artofproblemsolving.com/c/b/5/cb5e2a1efd462abd2354f4136f5fa64408f3138c.png)
