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 set of real numbers. Determine all functions 
that satisfy the equation![\[f(x^2f(x+y))=f(xyf(x))+xf(x)^2\]](http://latex.artofproblemsolving.com/b/4/b/b4b1e22e937394550f13cfdab142ca5bf4a26aea.png)
for all real numbers 
and 
.
+1 w
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
, prove that![\[ a = b\cos C + c\cos B. \]](http://latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
be reals such that 
. Prove that
Let be reals such that 
. Prove that
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.
such that
holds for all 
.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
be the set of the first perfect squares of nonzero integers. Suppose that 
for some 
. Here 
stands for the set 
. Prove that 
holds for every 
.
be a rectangle and 
any point. Let 
be the foots of the perpendiculars from to the lines 
, respectively. Let the perpendicular bisectors of 
and 
intersect at 
, and those of 
and 
intersect at 
. Prove that the lines 
and 
are parallel., let 
be the midpoint of 
, and 
the midpoint of 
. Suppose 
. Prove that![\[
2\angle AED-\angle DCB =180^\circ.
\]](http://latex.artofproblemsolving.com/b/d/0/bd01635e002970e841cef9dcb98b154ad6a7a202.png)
, let the incircle 
touch sides 
at 
, respectively. Let lie on the line through perpendicular to 
. Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment .
satisfies the equation![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then must be a rectangle.
, define![\[
\alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC|
\quad\text{and}\quad
\beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|.
\]](http://latex.artofproblemsolving.com/3/e/e/3eef0425824a79ba8615288dcfe2df79fca048b1.png)
Prove that![\[
AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta.
\]](http://latex.artofproblemsolving.com/d/4/a/d4adc5d510093da3520b26cbce172cf791b2847a.png)
lie on the arc of the circumcircle of 
(this arc does not contain ) such that 
. Let 
denotes the midpoint of the chord 
. Show that 
.
be the circumcenter of the . Then 
. We draw a normal line from the point 
at 
and let it intersect the circumcircle in the point
. Since 
we have that is a line of symmetry of 
. According to this,
. Now, from the triangle inequality
.
, so
. That is why 
.
and 
. Prove that
Where 
is equivalent to 
,
.
and the triangle inequality imply that the left hand side of the desired inequality is at least 
.