[/quote]
,
?
be reals such that 
and 
.
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 
points on a plane is 
,
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
points of every color there exists a 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
,
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)
.
integers are such that 
is prime and 
.
such that 
.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a 
,
,
is __
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
be a sequence defined by 
.
if 
.
Y by Adventure10, Mango247
Two points 
and 
are given on the plane. Let 
be some point equidistant from and . We construct a sequence of points 
, 
, 
, 
, 
, where 
is the center of the circle circumscribed around the triangle 
. At what position of the point :
a) the point
will fall in the midpoint of the segment 
(while and further members of the sequence are not defined),
b) does the point coincide with ?
and 
are given on the plane. Let 
be some point equidistant from and . We construct a sequence of points 
, 
, 
, 
, 
, where 
is the center of the circle circumscribed around the triangle 
. At what position of the point :a) the point
will fall in the midpoint of the segment 
(while and further members of the sequence are not defined),b) does the point
coincide with ?This post has been edited 3 times. Last edited by parmenides51, Dec 19, 2022, 12:13 PM
