Found problems: 310
Given is a prime number $p$. Prove that the number $$p \cdot (p^2 \cdot \frac{p^{p-1}-1}{p-1})!$$ is divisible by $$\prod_{i=1}^{p}(p^i)!.$$
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .
Let $f$ be a function from the set of rational numbers to the set of real numbers. Suppose that for all rational numbers $r$ and $s$, the expression $f(r + s) - f(r) - f(s)$ is an integer. Prove that there is a positive integer $q$ and an integer $p$ such that
\[
\Bigl\lvert f\Bigl(\frac{1}{q}\Bigr) - p \Bigr\rvert \le \frac{1}{2012} \, .
\]
For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$.
How many different residues modulo $1000$ do you get from $n!!$ for $n= 1, 3, 5, …$?
$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$
Find all solutions in positive integers to $(n+1)^k -1 = n!$
Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.
Does there exist a non-negative integer n, such that the first four digits of n! is 1993?
A positive integer $n$ satisfies the equation $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$?
$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }15$
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.
Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?
The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term?
$ \textbf{(A)}\ 60\qquad
\textbf{(B)}\ 75\qquad
\textbf{(C)}\ 120\qquad
\textbf{(D)}\ 225\qquad
\textbf{(E)}\ 315$
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\]
[i]2019 CCA Math Bonanza Team Round #4[/i]
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 positive integers $m$ and $n$ satisfying $2^n+n=m!$.
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?
Find all possible $\{ x_1,x_2,...x_n \}$ permutations of $ \{1,2,...,n \}$ so that when $1\le i \le n-2 $ then we have $x_i < x_{i+2}$ and when $1 \le i \le n-3$ then we have $x_i < x_{i+3}$ . Here $n \ge 4$.
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.
Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes.
For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^{k}$ in the expansion $(1+x+x^{2}+x^{3})^{n}$. Prove that
$Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}$.
[hide="hint"]
Think of $\binom{n}{j}$ as the number of ways you can pick the $x^{2}$ term in the expansion.[/hide]
Find the sum of $1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!$, where $n!=n(n-1)(n-2)\cdots2\cdot1$.
For any $ n\ge 2 $ natural, show that the following inequality holds:
$$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$
Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.
Prove that for any positive integer $n$ the following identity holds $\frac{(2n)!}{n!}= 2^n \cdot (2n - 1)!!$
Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square?
(S Tokarev)