of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
be prime numbers such that their sum isn't divisible by 
.
positive integer quadruples satisfy:
Y by Adventure10
Let 
be a triangle, 
is the center of the circle circumscribed around it, 
the diameter of this circle. It is known that the lines 
and 
are parallel. Prove that the triangle is isosceles.
(Andrey Mostovy)
be a triangle, 
is the center of the circle circumscribed around it, 
the diameter of this circle. It is known that the lines 
and 
are parallel. Prove that the triangle is isosceles.(Andrey Mostovy)
This post has been edited 1 time. Last edited by parmenides51, Oct 11, 2020, 10:34 PM
Reason: proposer added
Reason: proposer added
,
.
both contain 
,
and 
be a diameter. Then 
since 
is a rectangle and 
since 
is an isosceles trapezoid. Hence, 
and 
is isosceles.![[asy]
pair A, B, C, D, E, O;
A = dir(180); D = dir(0); C = dir(130); E = dir(310); B = dir(80); O = (0,0);
draw(circumcircle(A,B,C), dashed+lightblue); draw(A--C--B, blue); draw(E--D, blue); draw(A--D); draw(C--E); draw(B--D);
dot("$A$", A, dir(A)); dot("$B$", B, dir(B)); dot("$C$", C, dir(C)); dot("$D$", D, dir(D)); dot("$E$", E, dir(E)); dot("$O$", O, -dir(B));
[/asy]](http://latex.artofproblemsolving.com/1/1/4/114ba3bb38b5602a316d1bbfc09552d6690e910e.png)
,
then: 
so: 
