be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at 
.
and 
the midpoints of 
and 
.
and 
meet 
and 
at 
and 
,
,
bisects both angles 
,
,
.
and 
Prove that
be positive real numbers such that 
Find the minimum value of 
lies on the bisector of an acute angle with vertex 
,
be the projections of 
meets the sidelines at points 
and 
such that 
.
and the circle with center 
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 
Prove that
Where 
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 
of them has more than 
elements. Show that there's a member of 
different subsets 
.
of the set 
,
such that:
,
,
,
.
Y by
Let 
be a triangle with altitudes 
intersect at orthocenter 
. Let 
be an arbitrary point lies on 
. 
be the reflection point of 
on 
. 
intersects 
at 
. Prove that 
passes through the circumcenter of 
.
be a triangle with altitudes 
intersect at orthocenter 
. Let 
be an arbitrary point lies on 
. 
be the reflection point of 
on 
. 
intersects 
at 
. Prove that 
passes through the circumcenter of 
.
This post has been edited 1 time. Last edited by ken3k06, May 25, 2022, 1:38 AM
,
and 
are symmetric wrt 
,
is tangent to 
.
and 
.
to 
,
and 
lie on 
and 
respectively. Let 
be the reflection of 
,
are concyclic. From here, proving 
is easy.
