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)
,
and that line 
is not parallel to line 
.
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
![\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]](http://latex.artofproblemsolving.com/4/d/2/4d21982ba412c0bb4a7985798e81bc5b01ac0e43.png)
given that 
,
,
,
are nonnegative real numbers such that 
.
intersects line 
and 
.
.
satisfy 
?
such that 
is a multiple of 
?
where 
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?
Y by
Let 
a triangle and 
it´s circuncircle. Let 
a point in arc 
that not contain 
, diferent of 
and such that 
and are not parallel. Let 
the intersection of and and 
the circumcircle of triangle 
. Prove that the measure of 
does not depend of the choice of .
a triangle and 
it´s circuncircle. Let 
a point in arc 
that not contain 
, diferent of 
and such that 
and are not parallel. Let 
the intersection of and and 
the circumcircle of triangle 
. Prove that the measure of 
does not depend of the choice of .
,
