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
is equal to the oriented angle DCB.
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 
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 triangle has one side that is the space diagonal (or long diagonal) of a cube and the other two sides lie on the surface of the cube whose side has length 
.
1. What is the smallest perimeter that such a triangle can have?
2. What is the smallest area that such a triangle can have?
.1. What is the smallest perimeter that such a triangle can have?
2. What is the smallest area that such a triangle can have?
This post has been edited 1 time. Last edited by parmenides51, Jan 1, 2021, 11:57 AM
Reason: title
Reason: title
.
triangle, which has area 
.
right triangle with area 
