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 
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 Mango247
Diagonals of convex quadrilateral 
intersect at point 
. Points 
and 
are the midpoints of segments 
and 
, respectively. On the segments 
, 
, 
, 
points 
, 
, 
, 
respectively such that 
, 
.
The circumscribed circles of triangles 
and 
intersect at points 
and .
The circumscribed circles of triangles 
and 
intersect at points 
and .
Prove that the points
lie on the same circle.
intersect at point 
. Points 
and 
are the midpoints of segments 
and 
, respectively. On the segments 
, 
, 
, 
points 
, 
, 
, 
respectively such that 
, 
.
The circumscribed circles of triangles 
and 
intersect at points 
and . The circumscribed circles of triangles 
and 
intersect at points 
and .Prove that the points
lie on the same circle.This post has been edited 1 time. Last edited by parmenides51, Apr 24, 2022, 11:53 PM
