[/quote]
,
,
and 
are taken such that the point 
and 
.
and 
,
and 
are taken so that 
and 
.
.
and 
,
.
and 
be intersection points of lines 
and 
with circumcircle 
of triangle 
,
and 
intersection points of lines 
and 
with side 
.
,
with 
such that 
divides 
for all positive integers 
and 
with 
.
be a positive integer, and let 
denote the sum of the positive divisors of 
.
smallest positive integer relatively prime to 
-smooth if all of its prime factors are 
.
denote the number of 
.
be the least quadratic non-residue mod 
.
is a non-negative arithmetic function and
is convergent for some 
,
,
such that any graph with 
vertices and at least 
and 
,
be vertices. Define a directed graph 
as follows: for any integer point 
,
,
.
,![\( G[k, l] \)](http://latex.artofproblemsolving.com/6/3/a/63a5df1124231155f3240a3e21e24d293d0e8db3.png)
denote the subgraph of ![\([k, l]\)](http://latex.artofproblemsolving.com/2/9/f/29f08f190960a99d062fde7f2128bcb635657f36.png)
.
such that for any integer 
,![\( G[r, r + \alpha - 1] \)](http://latex.artofproblemsolving.com/5/1/a/51afcceb73daae97f8fc61456bf043b73b2689d7.png)
of ![\( G[r, r + \alpha] \)](http://latex.artofproblemsolving.com/8/8/7/887e55f5636b842e7b64c9c03982b0374cdf1c9e.png)
(i.e., how many connected components and what each component is like).
of a graph be partitioned into 
,
.
.
.
-
.
where no more than one circle is disjoint.
be a median in an acute-angled triangle 
is chosen on the line through 
tangent to the circumcircle of 
so that 
.
and 
.
lies on the line 
be a cyclic quadrilateral for which 
.
and 
externally on the sides 
and 
respectively.
meets 
at 
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
.
and let 
be the 
are concyclic.
Y by Adventure10, Mango247
A triple 
of positive integers is called quasi-Pythagorean if there exists a triangle with lengths of the sides 
and the angle opposite to the side 
equal to 
. Prove that if is a quasi-Pythagorean triple then has a prime divisor bigger than 
.
of positive integers is called quasi-Pythagorean if there exists a triangle with lengths of the sides 
and the angle opposite to the side 
equal to 
. Prove that if is a quasi-Pythagorean triple then has a prime divisor bigger than 
.
.
.
!
!
