[/quote]
,
and natural number 
,
(use in a problem for junior) to practice. Can anyone share to me?
are there such that 
and 
?
and 
.
be an acute angled scalene triangle with circumcircle 
.
respectively. Let 
and 
be points on segments 
and 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
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 
and 
are acute and 
.
,
.
Y by
Let 
be the center of a circle 
circumscribed around an acute-angled triangle 
be the midpoint of that arc 
of circle that does not contain point 
and 
be the point of intersection of the altitudes of triangle 
. Find the angle 
if 
.
be the center of a circle 
circumscribed around an acute-angled triangle 
be the midpoint of that arc 
of circle that does not contain point 
and 
be the point of intersection of the altitudes of triangle 
. Find the angle 
if 
.This post has been edited 1 time. Last edited by parmenides51, Dec 4, 2020, 6:41 PM
Reason: title typo
Reason: title typo
.
,
with radius 
.
and the reflection of 
.
and 
