[/quote]
,
(use in a problem for junior) to practice. Can anyone share to me?
be an acute angled scalene triangle with circumcircle 
.
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
and 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
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 
be midpoints of 
and 
be reflections of 
.
have the same centroid.
and 
are acute and 
.
,
.
.
,
,
.
.
at 
.
be the midpoint of 
,
meet 
.
at 
is tangent to 
Y by
Let 
be a triangle with two isogonal points 
and 
. Let 
be the projection of 
on 
, 
. 
is the projection of on 
. 
is the projection of on 
, 
is the projection of on 
, 
is the symmetric of 
wrt 
.
Prove that
passes through a fixed point when and vary.
be a triangle with two isogonal points 
and 
. Let 
be the projection of 
on 
, 
. 
is the projection of on 
. 
is the projection of on 
, 
is the projection of on 
, 
is the symmetric of 
wrt 
.Prove that
passes through a fixed point when and vary.This post has been edited 2 times. Last edited by parmenides51, Sep 19, 2021, 6:34 AM
