[/quote]
,
and natural number 
,
are there such that 
and 
?
and 
.
is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of 
be a square with 
side length and with center 
.
is drawn where point 
lies on line 
.
,
where 
is the number of times the number 
appears in the list. So, for example, for the list 
,
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of 
,
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist 
,
.
be midpoints of 
and 
be reflections of 
.
have the same centroid.
and 
are acute and 
.
,
.
.
,
be real numbers such that 
Prove that
Let 
Prove that
,
.
at 
.
be the midpoint of 
,
meet 
.
at 
is tangent to 
Y by
On the side 
of the triangle 
a point 
different from 
and 
is chosen so that the bisectors of the angles 
and 
intersect on the side 
. Let 
be the symmetrical point to with respect to the line . Prove that the points 
and are on the same line.
of the triangle 
a point 
different from 
and 
is chosen so that the bisectors of the angles 
and 
intersect on the side 
. Let 
be the symmetrical point to with respect to the line . Prove that the points 
and are on the same line.This post has been edited 2 times. Last edited by parmenides51, Dec 21, 2022, 9:11 PM
)
)
,
, however if you consider ![$F' \in [AB]$](http://latex.artofproblemsolving.com/b/4/2/b422105a4ce7941883ba86067bf31b080605b612.png)
the intersection of the parallel through 
you would quickly get 
,
and 
are bisectors it follows that 
,
,
.
