[/quote]
,
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.
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 
and 
and 
are on the same side of 
be midpoint of segment 
with 
and 
be positive real integer such that 
Prove that
such that 
and 
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
.
meets the side 
and 
are 
and 
and 
.
is a rectangle if and only if 
.
Y by
The Republic of Mathematics has decided to make a commemorative coin whose face side represents Gauss in an inscribed circle in a rectangle, itself inscribed in a more large circle of radius 
. Express the perimeter of the rectangle as a function of , if the area in gray at the outside of the small circle represents a fraction 
of the area of this small circle containing Gauss.
. Express the perimeter of the rectangle as a function of , if the area in gray at the outside of the small circle represents a fraction 
of the area of this small circle containing Gauss.This post has been edited 1 time. Last edited by parmenides51, Mar 13, 2021, 8:13 PM
