[/quote]
,
such that 
and 
.
.
and 
lie on the line segments 
,
.
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points 
and 
are concyclic.
,
,
with center 
and radius 
.
the midpoint of side 
intersects the circle 
.
be a point on the circle 
and 
.
.
.
and from 
to 
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
be all the divisors of 
.
are all divisors for some 
(
)
be a sequence of positive real numbers that satisfies 
and 
for every natural number 
for every natural number 
?
be positive real numbers such that : 
. Prove that : 
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
(ans- 
)
satisfying 
Prove that 
When does equality hold?
Y by Adventure10, Mango247
Let 
a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on 
, that
is tangent to the other three sides of the cuadrilateral.
(i) Show that
.
(ii) Calculate, in term of
and 
, the maximal area that can be reached for such quadrilateral.
a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on 
, thatis tangent to the other three sides of the cuadrilateral.
(i) Show that
.(ii) Calculate, in term of
and 
, the maximal area that can be reached for such quadrilateral.
then 
and ABCD is a rectangle with 
Assume that AC, BC are not parallel and they meet at a point P. Since ABCD is cyclic, 
and the triangles 
are similar. Let 
be the center of the semicircle tangent to AD, BC, CD. (E) is the excircle of the triangle 
against the vertex P with exradius 
Let (I) be the incircle of the triangle 
with inradius r. Denote 
the sides and 
the semiperimeter of the triangle 
Let the bisector 
of the angle 
meet the opposite side CD at a point K. The similarity coefficientb of the triangles 
is 
The incenter I cuts the angle bisector PK in the ratio 
hence 
Also, 
Thus the similarity coefficient is equal to
the center of the circle 
tangent to 
,
,
the respective points where 
,
.
and 
are obviously equal and rectangle at 
.
the radius of 
and 
.
,
,
,
,
are parallel, so are 
and 
,
(incidentally, 
)
be the point where they meet. Since 
and 
are similar, with proportionality constant 
,
times the area of 
and 
:
,
.
.
onto 
.
and the maximum area of 
I know one very nice proof of this part(without using trigonometry) 
If someone want i could post it.
Rays 
bisect 
and 
Pick the point 
on 
Then the triangle 
is isosceles, where
is cyclic 
we have
is then isosceles with apex 
