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.
are distinct and parallel, then
,
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
Seventh-grader Petryk played in the sand on the beach. He first drew an equilateral triangle 
. Subsequently he marked points 
and 
on the sides 
and 
, respectively, using small shells so that they divide the sides to which they belong, in a ratio of 
and 
, counting from the vertex 
. An unexpected wave washed the triangle out of the sand, but the shells remained in place. Help Petrik to reconstruct the triangle by performing geometric constructions with the help of a compass and a ruler.
(G. Filippovsky)
. Subsequently he marked points 
and 
on the sides 
and 
, respectively, using small shells so that they divide the sides to which they belong, in a ratio of 
and 
, counting from the vertex 
. An unexpected wave washed the triangle out of the sand, but the shells remained in place. Help Petrik to reconstruct the triangle by performing geometric constructions with the help of a compass and a ruler.(G. Filippovsky)
This post has been edited 1 time. Last edited by parmenides51, Jul 6, 2021, 3:08 PM
