.
Let 
.
. Prove that
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
and 
n
.
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](http://latex.artofproblemsolving.com/9/3/7/9377324cb6585daaf2cbec7fed6fce2209835e22.png)
where![\[a_k =
\begin{cases}
3^k, & \text{if } k \text{ is even}, \\
0, & \text{if } k \text{ is odd}.
\end{cases}
\]](http://latex.artofproblemsolving.com/8/b/e/8be520a30bdfd9272163470ee69120c2fd3e9b59.png)
Find the remainder when 
is divided by 
white balls, 
red balls and 
black balls. We extract balls at random without replacement until the bag becomes empty. What is the probability that there are still black and red balls in the bag when the last white ball is drawn?
Y by PikaPika999, nmlikesmath
Hi everyone,
I just have a question. I live in PA and I sadly didn't make it to nationals this year. Is PA a competitive state? I'm new into mathcounts and not sure
I just have a question. I live in PA and I sadly didn't make it to nationals this year. Is PA a competitive state? I'm new into mathcounts and not sure
.
