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with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
![\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]](http://latex.artofproblemsolving.com/4/d/2/4d21982ba412c0bb4a7985798e81bc5b01ac0e43.png)
given that 
, 
, 
, 
are nonnegative real numbers such that 
.
be the orthocenter of acute triangle , let be the foot of the altitude from 
to 
, and let be the reflection of across . Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
satisfy 
?
such that 
is a multiple of 
?
is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
where and are constants, 
is the time in minutes, 
is the length of the trail in miles, and 
is the altitude gain in feet. The model estimates that it will take 
minutes to hike to the top if a trail is 
miles long and ascends 
feet, as well as if a trail is 
miles long and ascends 
feet. How many minutes does the model estimate it will take to hike to the top if the trail is 
miles long and ascends 
feet?
distinct primes?
be the incenter and the circumcenter of triangle , and 
be the circumcenters of triangle 
. Let 
be the midpoints of segments 
. Prove that the circumcenter of triangle 
, , is the midpoint of segment 
.
be the incenter and the circumcenter of triangle , and be the circumcenters of triangle . Let be the midpoints of segments . Prove that the circumcenter of triangle , , is the midpoint of segment .
denote the circumcircle of triangle . Then it is well known that is the intersection of 
with and similar results hold for and .
with ratio 
. Then this sends to , to 
and to . Therefore it sends the circumcentre of 
, to the circumcentre of 
, . It immediately follows, then, that is the midpoint of , as desired.
are the midpoints of segments 
, respectively, the triangles and are similar, and since 
and 
are concurrent at , both these similar triangles also have the same orientation on the same plane. Therefore, 
and with being the midpoint of 
, is the midpoint of .
belongs to the perpendicular bisector of 
, the proof is pretty easy:
be the middle of 
.
, 
is miedial basis, so 
.
, so is the circumcenter of 
.
by angle chasing.Then it follows that 
by midpoint theorem.
lie on circumcircle of 
, so circumcentre of 
is . Applying Homothety with center and ratio 
, we get the result.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
Take a homothety at 
and note that 
so 
