[/quote]
,
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
are coprime. Consider the sequence 
given by 
.
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
be an integer and let 
be 
distinct subsets of 
.
such that the n subsets 
are also disjoint.
s
.
satisfies 
.
,
.
such that for all irrational numbers 
and 
,
where 
and 
.
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
with incenter 
,
touches sides 
and 
at points 
and 
,
and 
touches 
.
meets 
at 
.
is parallel to 
.
with incircle 
-
.
and 
Prove that
Let 
Prove that
Let 
Prove that
of positive integers such that ![\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]](http://latex.artofproblemsolving.com/3/5/3/353bbb15bac205053c26208af8cac7c32a296f55.png)
Proposed by Gabriel Chicas Reyes, El Salvador
Y by Adventure10
Each of 
numbers placed in a circular way is either 
or 
. (Both values appear). Consider the total sum of the products of every 
consecutive numbers.
Find the minimal possible value of this sum.
Find the maximal possible value of this sum.
numbers placed in a circular way is either 
or 
. (Both values appear). Consider the total sum of the products of every 
consecutive numbers.
Find the minimal possible value of this sum.
Find the maximal possible value of this sum.
,
.
should have same sign. Now let's look at the first block 
,
to be negative and in the first block.
it should be negative and the block that starts at 
.
should contain odd number of negatives, but 
,
are negative.
,
.
