such that:
a regular hexagon inscribed in a circle 
.
be a point inside 
its polar reflection wrt 
meet 
.
the polygon formed by the vertices 
using 
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
with an area of 
,
.
,
.
and 
,
,
.
define
(independent of 
and of 
)![$$
\lim_{x\to 1^{-}}
F_{a,\beta}^{(k,r)}(x)\,
e^{-\Lambda_{k,r,\lambda}\,(1-x)^{-\gamma}}
\;=\;
\begin{cases}
0, & a<c,\\[6pt]
\infty, & a>c.
\end{cases}
$$](http://latex.artofproblemsolving.com/e/3/8/e38c6b43d513bcc697587fff610a846b3a1b0e0a.png)
explicitly and verify that it coincides with the classical case 
,
.
(your answer may depend on 
but not on 
there exist nonnegative integers 
and integers 
such that 
be a field of characteristic 
,
.
is the square of an element from 
from 
be written in the same way?
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
or show that the limit does not exist.
be real numbers. Define 
and 
.
and hence find the limit.
![$f \in C^1([0,1]), g \in C^2(f([0;1]))$](http://latex.artofproblemsolving.com/f/5/2/f52369171ef8571734950120c43aacac10154fcd.png)
.
?
,
are positive integers and 
and 
.
with multiplication of matrixes operation is making an isomorphic-group structure with 
.
.
Y by internationalnick123456
Is it possible to color all lattice points in plane into 3 colors such that
1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).
2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
I think the answer is yes, but I couldn't find an example...
1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).
2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
I think the answer is yes, but I couldn't find an example...
This post has been edited 5 times. Last edited by CBMaster, Apr 11, 2025, 6:58 PM
Reason: .
Reason: .
