[/quote]
,
be positive integers and 
be prime. We know that 
and 
.
is a perfect square.
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 
.
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 PikaPika999, Fishheadtailbody
There are 
balls, with 
red, green, and blue balls, randomly arranged in a row. Two observers, one at the front and one at the back, each record the order of the first appearance of each color. What is the probability that both observers record the same order of colors?
For example, the sequence RGGBRB would be read as RGB for the front observer and BRG for the back observer.
balls, with 
red, green, and blue balls, randomly arranged in a row. Two observers, one at the front and one at the back, each record the order of the first appearance of each color. What is the probability that both observers record the same order of colors?For example, the sequence RGGBRB would be read as RGB for the front observer and BRG for the back observer.
