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 
friends for all 
. If there is no trio of students in this school, all three of whom are friends with each other, what is the minimum number of students in the school?
friends for all . If there is no trio of students in this school, all three of whom are friends with each other, what is the minimum number of students in the school?
in natural numbers m and n.
is an integer. Prove that 
are integers.
(mod 
)
