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 the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
and 
be arbitrary points on the sides 
and 
of triangle 
, respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point 
. Line 
meets line 
at point 
, while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
and 
Find the value of 
and 
such that 
and 
is a prime number
such that 
for all 
be an even integer and let 
be real numbers satisfying 
.
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy. such that 
is the cube of a positive integer. Find .
with respect to 
is greater than 0 if lies between and 
. (I've underlined it). But, I'm a little confused as I thought the power was 
and since is inside the circle, wouldn't the power be negative since 
?
.
, then you must double an odd number of times.
.
or 
, therefore three values work.
, then you must double an even number of times.
.
, or 
. Hence three values work.
.
thus, 
different dollar amounts.
