[/quote]
,
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 quadrilateral inscribed in a circle 
,
be a point on the extension of 
such that 
and 
are tangent to 
intersects 
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.
be tangent to the side 
.
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 Adventure10
On the sides of 
and 
of the rhombus 
, regular triangles 
and 
are constructed so that the point 
lies on the same side of as the line 
, and the point 
, on the other side of than 
. Prove that the points 
, and lie on the same line.
and 
of the rhombus 
, regular triangles 
and 
are constructed so that the point 
lies on the same side of as the line 
, and the point 
, on the other side of than 
. Prove that the points 
, and lie on the same line.
