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.
. Solve the equation: 
so that for any 
, we have ![\[ f(xf(x^2)+yf(x))=xf(y+xf(x)). \]](http://latex.artofproblemsolving.com/1/d/4/1d4fbf3f80306aaae5cfc11cc0e4be7ea41491d7.png)
, 
, such that
.
data values 
are all equal to 
. What is the value of ?
that are solutions of the equation 
.
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
plot 
equally spaced points around a circle. Label one of them 
, and place a marker at . One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of 
distinct moves available; two from each point. Let 
count the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that 
for all 
.
, where 
and 
. Prove that 
couldn't be expressed as product of two non-constant polynomials with integer coefficients.
satisfying the following two conditions:
such that, for all indices 
, 
.
and for any index , the number![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of .
is a pentagon whose vertices lie on circle 
where 
. Let 
and 
intersect at 
, 
meet 
at 
. 
is the midpoint of arc 
on , not containing 
. If 
holds, then what is the value of 
?
be the real solution of the system of equations 
, 
, 
. The greatest possible value of 
can be written in the form 
, where 
and 
are relatively prime positive integers. Find 
.
be six positive real numbers. Prove that
