[/quote]
,
is called stylish if the set 
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
is stylish because the set 
can be partitioned as 
,
,
.
that verify the following equation for all 
:
is a prime number
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
,
and 
although its supposed to be a quick comparison.
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
,
be a point on the extension of 
such that 
and 
are tangent to 
and the line 
at 
.
be the second point of intersection between 
and 
be a triangle with incenter 
,
and 
at 
.
and 
the reflections of 
with 
,
be the midpoint of 
and 
are perpendicular.
chance of landing heads when flipped. Krithik is playing a game where he starts with 
point. Every turn, he flips the coin, and if it lands heads, he gains 
points. Find the expected number of turns until Krithik wins.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
be the circle with diameter 
be the intersection of 
and 
be the intersection of 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
.
on 
on 
lies on the circumcircle of triangle 
.
be an isosceles triangle with 
Let 
Let 
intersect the extension of 
at 
Let 
and let 
and 
Let the extension of 
intersect 
and the circumcircle of 
respectively. Given that 
and 
compute the length of 
Y by Master_of_Aops, HWenslawski, son7
Three parallel lines 
are drawn in the plane such that the perpendicular distance between 
and 
is 
and the perpendicular distance between lines and 
is also . A square 
is constructed such that 
lies on , 
lies on and 
lies on . Find the area of the square.
are drawn in the plane such that the perpendicular distance between 
and 
is 
and the perpendicular distance between lines and 
is also . A square 
is constructed such that 
lies on , 
lies on and 
lies on . Find the area of the square.
,
,
,
and 
,
.
but i somehow calculated 
as 
instead of 
above 
,![\[ \text{Area}=a^2=6^2+3^2+45 \]](http://latex.artofproblemsolving.com/1/1/0/11005deb0fce6bacf495e0d89286f730b98a12b9.png)
angle between 
screams for complex come on! For sure unnecessary, but if first post can be trig, then this is fine as well.
for some real 
.
.
,
.
.
which is the desired answer. 
