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.
such that for all 
,![\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]](http://latex.artofproblemsolving.com/d/1/0/d10c01d7d416243614b24f04218d157e8b025ff2.png)
such that for all 
.
such that for all and 
such that 
there exist 
such that 
and 
. Note : 
and 
for all 
.
such that 
. Find the minimum value of:
such that for all . 
.
are colored blue. A set of six cells is called gremista when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of 
for which it is possible to color chessboard cells blue such that there is not a gremista set.
be non-negative real numbers such that 
Prove that
such that 
. Show that the equation 
has no integer solution.
denotes a polynomial of degree 
such that 
for 
, determine 
.
with 
, the foot of altitudes from 
to the sides 
are 
, respectively. 
is the orthocenter. 
is the midpoint of segment 
. Lines 
and 
intersect at 
. Let the tangents drawn to circumcircle 
from 
and 
intersect at 
. Prove that 
are colinear
and 
with real coefficients satisfying
lies on the circumscribed circle around 
, 
and 
are projections of point on 
and 
respectively. Prove that, if and 
are the middle points of and 
respectively, then 
is perpendicular to 
.
, 
. By Gliding principle 
is also similar to them. So 
.
is projection of point 
on 
and 
. 
and colinear, by Simson line.
-cyclic. 
, 
and 
, 
are median of 
, 
, respectively.
, so 
-cyclic. Therefore, 
lies on the circumscribed circle around , and are projections of point on and respectively. Prove that, if and are the middle points of and respectively, then is perpendicular to .
.
determine the minimum and maximum value of![\[\frac{\sum_{i,j=1}^n|a_i+a_j|}{\sum_{i=1}^n|a_i|}.\]](http://latex.artofproblemsolving.com/6/f/2/6f2427edf9a5e8e8f2e12276d0a841765e8322f1.png)
