[/quote]
,
is interesting if 2018 divides 
(the number of positive divisors of 
such that there exists an infinite arithmetic progression with common difference 
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers 
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
with the tangent to 
.
is tangent to 
,
denote the feet of the altitudes from 
,
.
and 
be the circles through 
,
other than 
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
is defined by 
,
for 
.
of ordered pairs 
for which there is a finite set 
of positive integers such that 
,
.
,
,
with the following property: An ordered pair of nonnegative integers ![\[m < \alpha x+\beta y < M\]](http://latex.artofproblemsolving.com/a/9/b/a9bb35658173fdab821cc8180420e47d5437e0a2.png)
if and only if 
.
.
,![\[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\]](http://latex.artofproblemsolving.com/0/9/b/09b8ed4e133ccf33b3f902fd9a9aa8e23c264341.png)
for all 
.
be 
positive integers such that the 
products![\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]](http://latex.artofproblemsolving.com/0/2/c/02cbff15c4d0118156a9d24be09360facb264d5a.png)
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
and, for each 
,
be the unique integer in the range 
for which 
is divisible by 
the obtained sequence is 
.
eventually becomes constant.
.
and 
has center 
is tangent to the circumcircle of triangle 
satisfying
Y by Adventure10, Mango247
The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat or a black hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case?
See also http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530553
See also http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530553
