[/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
There are 
lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?
lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?
and 
are not solutions.
:
:
:
)
:
at the end)
:
:
at the end)
for a lamp which is on and 
for a lamp which is off)
