.
.
?
since the coordinates correspond to the 6th roots of unity scaled by the radius. Since the sum of all n-th roots of unity is zero, the coordinates add up to zero.
,
inscribed in 
,
are respectively the midpoints of the major and minor arcs 
.
be the center of the inscribed circle, 
be 
,
is the intersection point of the line through 
parallel to 
intersects 
at 
,
on 
so that 
.
intersects 
,
intersects 
at 
.
is perpendicular to 
.
into what she calls the 
numbers, 
,
.
except that 
,
)
,
such that 
and both 
and 
.
For this to be equal to 
,
must be equal to 
,
,
,
,
and 
.![\begin{align*}
4&=[(t+ui)j]^4 \\
4&=(t+ui)^4 \\
4&=r^4(\cos4\theta+i\sin4\theta)
\end{align*}](http://latex.artofproblemsolving.com/9/3/e/93e59e5b32eea971180f54a729046aeaf0dbd599.png)
.
![\begin{align*}
4&=(\pm iB+ Bj)^4 \\
4&=[B(\pm i+j)]^4 \\
4&=B^4(-4)\\
-1&=B^4
\end{align*}](http://latex.artofproblemsolving.com/f/e/7/fe772c742c5f665ba9d8d42bc7a1066ebf4b6c9e.png)
and![\begin{align*}
4&=(A \pm Aij)^4 \\
4&=[A(1\pm ij)]^4 \\
4&=A^4(-4)\\
-1&=A^4
\end{align*}](http://latex.artofproblemsolving.com/9/f/a/9fa539c17f93e97ad28ba51a9785c9ae940cb800.png)
are the fourth roots of unity of 
t
.![\[
f(x) = \frac{1}{\sqrt[3]{x^2+2x+1}+\sqrt[3]{x^2-1}+\sqrt[3]{x^2-2x+1}}
\]](http://latex.artofproblemsolving.com/2/0/8/20806cb4860ace2b69aa76ea0a429a283ec76637.png)
for all positive integers 
,![\[
f(1)+f(3)+f(5)+\dots +f(7999)
\]](http://latex.artofproblemsolving.com/9/0/b/90bd38f627e03a496e81078a5944067084963afa.png)
be a triangle such that 
.
lie on segments 
,
and 
are the angle bisectors of triangle 
.
.
,![\[
S=\left\{ x \in \mathbb{Z} | \frac{x^2 - 2x +7}{2x-1} \in \mathbb{Z} \right\}
\]](http://latex.artofproblemsolving.com/d/0/6/d0681591c7ca805f605caafb20be9ff10e14ca06.png)
Hence find the number of elements in 
.
rolls have been made. If Alice makes the first roll, the probability that Bob makes the last roll can be written as 
,
and 
are relatively prime positive integers. What is 
?
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
Y by RainbowJessa, cubres, At777
I believe discussion is allowed after yesterday at midnight, correct?
If so, I will put tentative answers on this thread.
By the way, does anyone know the answer to Geometry Problem 5? I was wondering if I got that one right
Also, if you put answers, please put it in a hide tag
If so, I will put tentative answers on this thread.
By the way, does anyone know the answer to Geometry Problem 5? I was wondering if I got that one right
Also, if you put answers, please put it in a hide tag
This post has been edited 21 times. Last edited by Aaronjudgeisgoat, Apr 22, 2025, 11:32 AM
minutes to solve it.
game using the integers 
exactly once"
