AoPS Academy
?
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
. Each step the pig has probability 
of moving 
unit in the positive direction and probability 
of moving unit in the negative direction. Find the expected number of steps until the yellow pig visits 
.
, the sum of the products of the elements taken 2 at a time is 
; and the sum of the products of the elements taken 3 at a time is 
. Let 
be the sum of the products of the first 2000 positive integers, taken n at a time. For example, 
= the sum of the first 2000 positive integers, and 
= the sum of the products of the 2000 positive integers, taken two at a time, etc. What is the value of ++
+..........+
+
?
be positive integers such that 
. Find 
of 
is finite (and of course is an infinite set since 
for 
). But then, by Kobayashi, the set of prime divisors of 
is infinite. But 
, whose prime divisors must belong to the original set, a contradiction.
, if 
then 
. Therefore, we must get a new prime divisor for each 
, proving the result. 
, but that's easy.)
in natural numbers m and n.
is an integer. Prove that 
are integers.
(mod 
)
is defined by 
and 
Show that there exist infinitely many prime number that divide at least one number in this sequences
