[/quote]
,
be a triangle with circumcenter 
.
(see figure) and 
be the intersections of the perpendicular bisectors of 
and 
.
and 
and the point 
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to 
satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
side length and with center 
.
is drawn where point 
lies on line 
.
be an odd integer, 
.
integers 
such that for any 
,
and 
.
for which satisfy 
for some 
,
is greater than 
.
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
, such that 
Prove that :
equality cases : ?
distinct positive integers such that adding the product of any two of them to 
yields a perfect square?
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
is a prime less than 
for some positive integer 
,
such that 
.
Y by
Let 
be a cyclic quadrilateral such that dioganals 
and 
intersect at 
.Circumcircles of triangles 
and 
intersect at 
.Let 
and 
be the midpoints of 
and 
respectively.Circumcircle of 
intersects 
and 
at 
and 
respectively.Prove that 
and 
intersects at .
by Anar Abbas (Analgin)
be a cyclic quadrilateral such that dioganals 
and 
intersect at 
.Circumcircles of triangles 
and 
intersect at 
.Let 
and 
be the midpoints of 
and 
respectively.Circumcircle of 
intersects 
and 
at 
and 
respectively.Prove that 
and 
intersects at .by Anar Abbas (Analgin)
Attachments:
This post has been edited 2 times. Last edited by parmenides51, Apr 30, 2021, 8:32 AM
.
and denote the inverse of point 
by 
, then clearly 
and 
.
