AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
+1 w
be a function such that for every regular 
-gon 
we have 
. Prove that 
for all reals 
.
be positive integers, where 
. It is given that there exist a positive integer 
such that
. Prove that the above equation is true for all positive integers 
. (Here 
is the sum of digits of 
taken in base 
).
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
, there exist pairwise different real numbers 
satisfying
be a positive integer, and let 
denote all positive divisors of , If the following conditions are satisfied:
.
o 
cyclic we need that 
and 
are both beyond 
(with respect to 
).
and is cyclic 
.
.
and 
and 
.
.
is cyclic.
such that 
.we know that 
is cyclic and 
is cyclic . If 
then applying Reim to ![$ (KLB'C'),(ALB') [=ABC]$](http://latex.artofproblemsolving.com/1/2/d/12df27435c81cfb359ee1ea4022793f4ad28f064.png)
yields 
which is the tangent to 
at 
. let 
point of 
in the same semi-plane bounded by 
as 
.
thus 
then 
are colinear . cyclic 
from 
cyclic we deduce 
thus
a triangle, let 
be the foot of the bisector relative to and 
be the foot of the trisectrix relative to closer to the side 
(
). Let and be two point on the line 
on the side of with respect to , such that 
and 
. Prove that is cyclic if and only if lines 
and 
are parallel.
such that 
where 
and 
Prove that
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](http://latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
,
is the binomial coefficient 
choose 
.
