[/quote]
,
and 
Find the minimum 
are four given points on the plane, so that no three of them are collinear. Let 
be the set of conics passing through 
,
,
,
.
be the line joining the intersection of 
and 
with the intersection of 
and 
.
varies."
such that:![\[f(1+xf(y))=yf(x+y)\]](http://latex.artofproblemsolving.com/f/2/a/f2a9e8eb184c921d5f1fa59137d99e38b29c6962.png)
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
be an acute scalene triangle 
be the orthogonal projection of 
on 
,
and 
are the midpoints of 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
,
,
are positive integers that satisfy
and 
.
be real positive numbers such that: 
.
:
such that:
for all x,y E 
,
,
,
be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, 
.
for all 
.
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
,
.
intersects 
.
is orthocenter of triangle 
.
coincide
be integer numbers 
satisfy that
.
is a perfect square number
can’t be any power 
of a positive integer number
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
Y by Adventure10, Mango247
Let 
be the intersection of the diagonals of the cyclic quadrilateral 
. Let 
and 
denote the respective midpoints of the sides 
and 
. Prove that the line through which is perpendicular to 
, the line through which is perpendicular to 
and the line through which is perpendicular to 
are concurrent.
be the intersection of the diagonals of the cyclic quadrilateral 
. Let 
and 
denote the respective midpoints of the sides 
and 
. Prove that the line through which is perpendicular to 
, the line through which is perpendicular to 
and the line through which is perpendicular to 
are concurrent.
is sufficient (so that's a more general condition). Am I right? (I'll post my solution later.)
be the foot of the perpendicular from 
and 
intersect at 
.
and 
are similar. Hence we have (using similarity of triangles) that 
.
.
is cyclic, we get 
.
is the foot of the perpendicular on 
is the foot of the perpendicular on 
and 
cross at 
is cyclic and angle chasing to get 
showing 
collinear.
was collinear to 
with 
.
are collinear. Notice that it would be enough to prove that 
is a cyclic quadrilateral since then 
as the both subtend the same arc and therefore 
which implies that 
and therefore 
is cyclic. Thus, by the Power of a Point theorem we have that
and 
are similar. Moreover, since 
and 
are similar. From this we see that 
but we also know that 
since they subtend the same arc. Therefore, 
which shows that 
is cyclic. Using the Power of a Point theorem again we have that
which we combine with the first equation to get
which shows that 
.
