AoPS Academy
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
be a positive integer. Prove that there exist three permutations 
, 
, and 
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
, no matter how its elements are partitioned into two subsets, at least one of the subsets must contain three numbers 
(
is allowed) such that 
. Find the minimal .
and a point 
, let 
be the cevian triangle of with respect to . Let 
be the orthocenter of , and denote the isotomic conjugate of 
with respect to by 
, respectively. Let the centroid of be 
, and denote the isogonal conjugate of with respect to by 
. Prove that
or in brief
be prime numbers such that their sum isn't divisible by 
. Find the all 
positive integer quadruples satisfy:
is a positive integer, then 
divides 
. Now, see that the 
which is 
or 
; but, as divides 
then divides .
then is clear than 
, 
, 
and is even, or 
, 
, and positive integer.
, then 
or ; otherwise, and 
are and 
and divides . So, if and 
, sp 
. So 
and 
, 
and 
or 
and 
. If and 
then and .
.
and 
for every 
-
:
