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 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at 
. Denote by 
and 
the midpoints of 
and 
. Rays 
and 
meet 
and 
at 
and 
, respectively. Prove that there exists a point 
, lying outside quadrilateral , such that
bisects both angles 
, 
, and
.
be positive real numbers such that 
Find the minimum value of 
lies on the bisector of an acute angle with vertex . Let 
, 
be the projections of to the sidelines of the angle. The circle centered at with radius 
meets the sidelines at points 
and 
such that 
. Prove that the circle with center touching 
and the circle with center touching 
are tangent.
are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing .
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
be a set of 
elements. Let 
be subsets of such that the union of any 
of them has more than 
elements. Show that there's a member of that occurs in at least 
different subsets 
.
of the set 
, define its colorfulness as the greatest natural number 
such that:
, 
, if 
, then 
.? Determine how many such permutations have maximal colorfulness.
be a triangle with incenter 
and circumcenter 
. the orthocenter of triangle 
. The incircle of touches side 
at 
respectively. Suppose that 
and 
intersects 
at and respectively (differ from ). Prove that 
lies on a circle. denote the circumcircle of )
about the perpendicular at onto 
(i.e. that line is the locus of 
's circumcenter); I am not able to provide more info as yet!
through !
) is actually the reflection of point about the perpendicular to at the point 
given by 
, or 
. be the center of the circumcircle of 
(of radius 
) and the foot of the perpendicular from to . Then the given expression is equivalent to 
or (using the power of and around circle ) 
, or 
. That is, is a fixed, given point on . at . This perpendicular bisects the common chords 
of all circumcircles , and thus is the reflection of about this perpendicular.
is on 
so that 
, then 
is one limiting case of with 
and 
. Indeed, in this case, 
and 
. On the other hand, in the limiting case of 
, we get 
. To make sure that this is possible, we note that the previous quadratic in 
has a solution if 
, which is allowed by the given condition that 
.
with arbitrary radius sends 
to line 
and 
to points on lines 
such that 
Setting 
, we have 
for some constants 
(using 
, etc). passes through a fixed point different from iff line 
goes through a fixed point. Take an affine transformation sending to perpendicular axes 
and 
respectively. Then it is straightforward to check that line 
(the image of 
after the transformation) goes through the fixed point 
, as desired.
is fixed, not necessarily equal to 1.
Prove that
and 
Prove that
Prove that
Where 
.
.
let 
such that 
.
but 
so 
and now 
so 
passes through fixed point 
which is equivalent to 
or (if 
)
or 
,
and 
.
,
meets 
,
)
,
.
,
,
same solution as leader