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 a solution then so is 
for any constant 
so assume 
. By condition divides all the numbers 
so 
. Let 
where g is bounded since 
now swapping x,y with y,x we see that 
meaning 
but note that is strictly increasing and 
is bounded so is eventually . Fix some t ,, for large 
meaning 
but since and is increasing by induction 
for all so 
for any 
.
.
, then 
. By induction it is clear that if we assume 
, then putting 
gives us 
.
. Then because of Claim 1, we have . is clearly incereasing and putting 
gives 
. Combining with the fact 
gives us that 
after some point. Therefore, 
for some constant 
and sufficiently large 
. If 
, taking and 
gives a clear contradiction. is increasing thus 
for all .
where 
is a positive integer.
in natural numbers m and n.
is an integer. Prove that 
are integers.
(mod 
)
such that
is again a positive integer not exceeding 
.
.
,
,
is