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.
+2 w
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 
.
players. A round involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called balanced if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of 
rounds. Prove that the committee can organise a tournament this year with balanced rounds.
and 
to the circumcircle of an acute angled triangle 
meet the tangent at 
at 
and 
respectively. 
meets 
at 
, and 
is the midpoint of 
; 
meets 
at 
, and 
is the midpoint of 
. Prove that 
. Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
satisfying
Prove
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
satisfying![\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]](http://latex.artofproblemsolving.com/c/6/2/c6237d13efcb3f2ef74cfdbede1a6583ce970e79.png)
For all 
.
+1 w
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
questions.
be a point in the interior of 
such that 
, 
, 
, 
, 
, 
. Show that 
.
and 
are the points of intersection between the circumcircle of ABC and EF. 
, 
. Show that 
are concyclic with centre in A.
instead of 
.
is closer to 
and 
is to 
. The key points of the proof are the following:
lie on a circle with center 
.
Prove that
Where 
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
,![\[\frac{2}{3}\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right)\leq\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+d}+\sqrt{d+a}\leq 2(\sqrt{2}-1)\left(\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\right).\]](http://latex.artofproblemsolving.com/2/d/4/2d48c32aff8f3b3992202f149d835e4ee0982c1b.png)
