Found problems: 310
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.
[i]Proposed by Michael Kural[/i]
Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.
Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.
Find all the pairs of natural numbers $(a, b),$ such that
\[a!+1=(a+1)^{(2^b)}\]
Let $n!=ab^2$ where $a$ is free from squares. Prove, that for every $\epsilon>0$ for every big enough $n$ it is true, that $$2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}$$
[i]M. Ivanov[/i]
Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.
Find roots of the equation
$$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.