[/quote]
,
such that 
.
Where 
be an equilateral triangle. Let 
be a point on 
be a point on 
,
be a point on 
such that 
and̸ 
.
and 
,
.
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of 
and 
Prove that
Prove that
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
(possibly 
)
for every integer 
.
of nonzero integers such that the only admissible set containing both 
and 
is the set of all integers.
be the circumcenter and orthocenter of triangle 
and 
intersect at point 
.
and 
be the intersection of 
w
, respectively.
are concyclic.
with incenter 
.
and 
intersects 
and 
and 
be the midpoints of 
meets the external bisector of angle 
and line 
meets the external bisector of angle 
.
are concylic.
Y by Mango247, Mango247, Mango247
A mountain is of a perfectly conical shape. The base is a circle of radius 
miles, and the steepest slopes leading up to the top are 
miles long. From a point 
at the southernmost point of the base a path leads up on the side of the mountain to 
, a point on the northern slope and 
of the way to the top, (i. e. if 
is the top and 
the northernmost slope, 
being on the ground, then 
).
If
is the shortest path leading along the mountainside from to , find
(i) the length of the whole path,
(ii) The length of the part of the path between
and where is a point on the path where it is horizontal.
miles, and the steepest slopes leading up to the top are 
miles long. From a point 
at the southernmost point of the base a path leads up on the side of the mountain to 
, a point on the northern slope and 
of the way to the top, (i. e. if 
is the top and 
the northernmost slope, 
being on the ground, then 
).If
is the shortest path leading along the mountainside from to , find(i) the length of the whole path
,(ii) The length of the part of the path between
and where is a point on the path where it is horizontal.
(on the edges) and the path from 
,
by power of point. So chord 
.
