$p$ and $q$ are distinct prime numbers. Prove that the number \[\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}\] is an integer.

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 all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

Let \[ S = \sum_{i = 1}^{2012} i!. \] The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$. [i]Proposed by Lewis Chen[/i]

Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.

Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.

If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end? (With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).

Find all natural numbers such that each is equal to the sum of the factorials of its digits. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

The product $1 \cdot 2 \cdot \ldots \cdot n$ is denoted by $n!$ and called [i]n-factorial[/i]. Prove that the product \[ 1!2!3!\ldots 49!51! \ldots 100! \] (the factor $50!$ is missing) is the square of an integer number.

Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.

For integers $n \ge k \ge 0$ we define the [i]bibinomial coefficient[/i] $\left( \binom{n}{k} \right)$ by \[ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .\] Determine all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer. [i]Remark: The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

An integer $n \geq 3$ is called [i]special[/i] if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$. Find all special numbers $n$ such that $10 \leq n \leq 100$.

Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]

Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

Does there exist a non-negative integer n, such that the first four digits of n! is 1993?

In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below. \[\begin{array}{c@{\hspace{8em}} c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{6pt}} c@{\hspace{6pt}}c@{\hspace{6pt}}c} \vspace{4pt} \text{Row 0: } & & & & & & & 1 & & & & & & \\\vspace{4pt} \text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\vspace{4pt} \text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\vspace{4pt} \text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\vspace{4pt} \text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\vspace{4pt} \text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\vspace{4pt} \text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1 \end{array}\] In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?

At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?

Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$.