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.![$P(x) \in \mathbb{Q}[x]$](http://latex.artofproblemsolving.com/9/1/0/9104130500a15ab4bffb68bc493faa4ea2891610.png)
be a polynomial with rational coefficients and degree 
. Prove there is no infinite sequence 
of rational numbers such that 
for all 
.
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
aand 
Prove that
Let 
aand Prove that
Let 
aand 
Prove that 
with the center 
touches the sides 
at 
respectively. The line 
meets the circumcircle of at 
. The line 
meets the line 
at 
and the line 
meets the circumcircle of at 
. Define 
similarly. Prove that the lines 
are concurrent.
intersect at point 
. Point 
lies between the parallel lines 
and 
such that 
, and line 
separates points and . Prove that 
.
and 
Prove that 
be an even integer and let 
be real numbers satisfying 
.
be a triangle with incentre and circumcircle 
. Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.
be the circumcircle of 
and let 
be A-excircle .Let the two common tangents of 
cut in 
.Prove that 
.
be an acute triangle with circumcenter 
and orthocenter 
, satisfying 
. The tangent line at 
to the circumcicle of intersects in 
. Let 
be the midpoint of 
. Prove that 
.
be midpoint of side . It is well-known that 
is parallelogram. is orthocenter of triangle 
. Since 
and 
, we deduce that 
. 
and triangle 
, where is the midpoint of side .
aand 
Prove that 
aand 
Prove that
Let 
aand 
be an integer greater than or equal to 
.
satisfy 
,![\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]](http://latex.artofproblemsolving.com/2/b/b/2bbca5c72fe6bf9f3fa50cc34b8bff27f2ec5d4d.png)
must hold.
,
are 
real numbers satisfying 
,![\[ \sum_{1\leqslant i < j\leqslant n}\frac {1}{1 - x_ix_j}\leqslant\frac {n^2}{2}\mbox{.}\]](http://latex.artofproblemsolving.com/7/9/e/79e8a348888aed07bbd757ffbf01e85f33d5ada0.png)
