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.
satisfying![\[
f(x^2 + y^2) = f(x - y)f(x + y) + \alpha y f(y), \quad \forall x, y \in \mathbb{R}
\]](http://latex.artofproblemsolving.com/d/2/a/d2aedbc5ef35cfceed8fe9a51771e3caa449d7bb.png)
(where 
is a fixed real constant).
, define the sequence 
for 
as
Determine all values of 
such that there exists a number 
such that 
for infinitely many values of 
.
denote the set of positive real numbers. Find all functions 
such that for all positive real numbers 
and 
, ![\[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]](http://latex.artofproblemsolving.com/9/c/a/9ca75afc75318aedde5d1f26ef84e4bebfc90d8c.png)
of positive integers such that 
is divisible by 
.
be three diameters of the circumcircle of an acute triangle 
. Let 
be an arbitrary point in the interior of 
, and let 
be the orthogonal projection of on 
, respectively. Let 
be the point such that 
is the midpoint of 
, let 
be the point such that 
is the midpoint of 
, and similarly let 
be the point such that 
is the midpoint of 
. Prove that triangle 
is similar to triangle .
be a non-isosceles triangle with incircle (
). Denote by 
the points where () touches 
, respectively. The A-excircle of is tangent to 
at 
. The lines 
and 
meet 
at 
and 
, respectively.
, 
, and 
are concurrent at a point.
and 
be the midpoints of and , respectively, and let 
be the orthocenter of triangle 
. Show that the points 
are collinear.
be a triangle and let and 
be points on the segments 
and 
respectively such that 
. Let 
be the intersection point of segments 
and 
and let 
be the second intersection point of the circumcircles of 
and 
. Prove that lies on the bisector of the angle 
.
for which ![$\dfrac{n^2+1}{[\sqrt{n}]^2+2}$](http://latex.artofproblemsolving.com/7/e/d/7ed6c67114c0c7d262f67e5b7a733daab7997415.png)
is an integer. Here ![$[r]$](http://latex.artofproblemsolving.com/2/f/c/2fcabe2c2b8ebe2c428830acb9874cb8a8df790f.png)
denotes the greatest integer less than or equal to 
.
make the equation ![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
has three distinct real roots 
, 
and 
. The polynomial 
has roots ![$\sqrt[3]{\alpha}$](http://latex.artofproblemsolving.com/4/f/5/4f57cc54c79420c509385c423ffa42820cef3109.png)
, ![$\sqrt[3]{\beta}$](http://latex.artofproblemsolving.com/d/4/0/d40f4039c320468d9974a9504917b56eaf73c947.png)
, ![$\sqrt[3]{\gamma}$](http://latex.artofproblemsolving.com/7/8/1/78171ba6cd442d8156fd70b3d845075e3d82a926.png)
. Find the integer closest to 
.
is the largest value of the constant 
such that ![\[{ab+ac+ad+bc+bd+cd-6}\leqslant{k{\left(a+b+c+d-1\right)}{\left(a+b+c+d-4\right)}}\]](http://latex.artofproblemsolving.com/b/e/7/be787de190655249cbfa6db6705a0d614231ad18.png)
holds for any nonnegative real numbers 
satisfying 
.
and 
Prove that
.