AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be reals such that 
and 
Prove that
Let be reals such that 
and Prove that![$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$](http://latex.artofproblemsolving.com/6/2/2/62211222c148c86d65526d86dde603cb6598925f.png)
![$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$](http://latex.artofproblemsolving.com/3/8/3/383d38cc87c1b1ea1e17e10e4bb95bed79329786.png)
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
. The tournament adopts an elimination system, and the winner advances to the next round. What is the probability that the Argentine team will meet the Brazilian team in the entire tournament?
. 
. 
. 
. 
. 
Prove that
Let 
Prove that
Let 
Prove that
distinct points in a plane such the circumradius of any 
of them is the same.
). We have 
distribute the 
factors of 
among the values. Continuing for all primes gives the product 
. For the example, the answer is 
. The number of unordered pairs would be messy because you have to do casework with pairs containing repeated values.
, how many pairs of numbers multiply to 
Prove that
Where 
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
with prime factorization 
,
numbers that multiply to 
,
,
then their product far exceed 
)
I think?