[/quote]
,
are positive real numbers satisfied 
.
is written on the blackboard. Every time, Carmela can erase the number 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
be a square with sides length 
.
be a path within 
with 
.
on the boundary of 
.
and 
of 
and the length of the part of 
.
and the sets 
If 
has exactly three elements, prove that 
cannot be an integer.
and 
are divisible by 
is the sum of digits of 
.
other books each in one copy?
.
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
so that the sum of its edges' lengths is minumum.
find, in terms of 
for which there is a set of 
distinct positive integers that has sum greater than 
with 
equilateral. Let 
,
and 
.
where 
is the circumradius and 
is the volume of the tetrahedron.
,
be the orthocenter of 
.
at 
.
at 
be the midpoint of 
,
meet 
at 
.
is tangent to 
where 
.
we have 
.
there exists 
such that![\[u<P(n)-\lfloor P(n)\rfloor <v.\]](http://latex.artofproblemsolving.com/9/4/a/94a8cb1f064e1dea37e8894fb4fd4e34c4c6c845.png)
Generalize the problem!
which moves on line 
towards 
.
.
,
.
Y by
For each pair of positive integers 
and 
, we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.
and 
, we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.This post has been edited 2 times. Last edited by EmersonSoriano, Yesterday at 5:25 PM
Reason: change
Reason: change
.
is prime hence 
.
then 
hence 
where 
is another prime. If 
then 
.
on the left for 
.
.
.
then 
hence 
.
then 
and we already have terms 
on the LHS for 
.
and 
gives the remaining sum 4. Hence 
with 
we find 
with 
then 
.
on the LHS hence nothing more and 
.
