[/quote]
,
such that 
but 
does not divide 
.
and ![\[ 3n^2+3n+7=m^3 \]](http://latex.artofproblemsolving.com/6/f/8/6f84f81f857094b3ea432284a7881697167b2418.png)
be a point on the Neuberg cubic. Show that as 
with the property that 
is divisible by 
.
be the circumcircle of a non-isosceles triangle 
and 
be a tangent line to 
)
be midpoint of 
and 
.
and 
be coprime positive integers such that:![\[ \dfrac{p}{q}=1-\frac12+\frac13-\frac14 \cdots -\dfrac{1}{1334}+\dfrac{1}{1335} \]](http://latex.artofproblemsolving.com/b/d/3/bd374388fe6b84fbcd66e23a7956f2fbeebe4c28.png)
and 
satisfying 
and 
.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
(the number of positive divisors of 
such that there exists an infinite arithmetic progression with common difference 
,
.
that satisfies:
satisfy 
and 
Prove that always exists positive integer 
.
is "not congruence"
Y by Adventure10, Mango247
Let 
be a triangle and the point 
is on the segment 
such that 
is the interior bisector of 
. We stretch such that it meets the circumcircle of 
at 
. We draw a line from such that it meets the lines 
at 
, respectively ( is not between 
and also is not between 
).
Prove that
.
be a triangle and the point 
is on the segment 
such that 
is the interior bisector of 
. We stretch such that it meets the circumcircle of 
at 
. We draw a line from such that it meets the lines 
at 
, respectively ( is not between 
and also is not between 
).Prove that
.
beyond 
and 
lies on segment 
.
,
.
lies outside the circle 
.
and 
.
,
.
.
and 
.
,
,
,
,
.
it means that 
(1)
is tangent to 
(2)
and then by angle bisector theorem get that : 
and by sines Theorem get that 
and 
degrees and then finish the problem by using 
