[/quote]
,
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
and 
,
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
intersect at points 
and 
.
and 
.
,
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
?
and 
points on a plane is 
,
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
,
be the set of all positive integers not exceeding ![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
Y by Adventure10, Mango247
Let 
and 
be points on the sides 
and 
, respectively of the parallelogram 
, such that 
. Let 
be the intersection point of 
and 
. Show that 
is bisector of angle 
.
and 
be points on the sides 
and 
, respectively of the parallelogram 
, such that 
. Let 
be the intersection point of 
and 
. Show that 
is bisector of angle 
.This post has been edited 2 times. Last edited by parmenides51, Jul 23, 2021, 9:23 PM
Reason: title
Reason: title
is not parallel to 
so extend 
.
and again by similarity 
.
.
.
