probability of coming up heads and 
probability of coming up tails. The expected number of flips necessary to first see the sequence 
in that consecutive order can be written as 
for relatively prime positive integers 
,
.
be four distinct points in the plane.
and 
are distinct and parallel, then
lies on the segment 
,
plus the oriented angle 
e
.
equals 0°, then
a
and 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
be reals such that 
. Prove that
specifies the greatest common divisor of 
and 
.
and 
Prove that
Let 
Prove that
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
cannot completely cover a square of side length 
,
,
.
to meet the circumcircle of 
at 
,
.
Y by
A group of oceanographic researchers has three platforms, 
, and 
, on the Pacific Ocean, each containing a laboratory. Their research is limited to the portion of the ocean included in the spherical triangle 
. Knowing that 
, 
, 
, determine the total area of the spherical triangle (in terms of the area of the earth).
Note: Unlike the case of plane triangles, in a triangle spherical, the sum of the angles is not equal to
degrees. She can vary by always being greater than degrees, as Illustrated in the following figure where the three angles are 
degrees each and their sum is 
degrees 
degrees.
, and 
, on the Pacific Ocean, each containing a laboratory. Their research is limited to the portion of the ocean included in the spherical triangle 
. Knowing that 
, 
, 
, determine the total area of the spherical triangle (in terms of the area of the earth).Note: Unlike the case of plane triangles, in a triangle spherical, the sum of the angles is not equal to
degrees. She can vary by always being greater than degrees, as Illustrated in the following figure where the three angles are 
degrees each and their sum is 
degrees 
degrees.This post has been edited 1 time. Last edited by parmenides51, Mar 13, 2021, 6:23 AM
.
,
of the earth.
