and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
be positive real numbers. Prove that 
.
,
integers and 
is a positive integer. Suppose that the integers
satisfy: 
are divisible by k. Prove that:
is divisible by k
denote the increasing arithmetic sequence of integers whose first term is 
and whose common difference is 
is the sequence 
.
?
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in 
satisfy![\[
a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015}
\]](http://latex.artofproblemsolving.com/8/c/0/8c0b0cfdd569ea250e97590a3364794920d86e04.png)
and 
.![\[
\frac{ab}{2015} - a + b.
\]](http://latex.artofproblemsolving.com/1/f/6/1f620dea7a0c769137bf13004916e8b283e00eee.png)
,
.
,
is a regular tetrahedron.
with respect to a circle 
,
,
,
Y by
Say that on the AMC 10, you do better on the A than the B, but you still qualify for AIME thru both. Then after your AIME, it turns out that you didn’t make JMO through the A+AIME index but you did pass the threshold for the B+AIME index.
does MAA consider your B+AIME index over the A+AIME index and consider you a JMO qualifier even tho Your A test score was higher?
does MAA consider your B+AIME index over the A+AIME index and consider you a JMO qualifier even tho Your A test score was higher?
