Found problems: 310
2019 CCA Math Bonanza, I8
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$, what are all possible values of $a$?
[i]2019 CCA Math Bonanza Individual Round #8[/i]
2000 Singapore Senior Math Olympiad, 2
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.
1998 Estonia National Olympiad, 4
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .
2016 Brazil Team Selection Test, 2
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
2014 Online Math Open Problems, 25
Kevin has a set $S$ of $2014$ points scattered on an infinitely large planar gameboard. Because he is bored, he asks Ashley to evaluate \[ x = 4f_4 + 6f_6 + 8f_8 + 10f_{10} + \cdots \] while he evaluates \[ y = 3f_3 + 5f_5+7f_7+9f_9 + \cdots, \] where $f_k$ denotes the number of convex $k$-gons whose vertices lie in $S$ but none of whose interior points lie in $S$.
However, since Kevin wishes to one-up everything that Ashley does, he secretly positions the points so that $y-x$ is as large as possible, but in order to avoid suspicion, he makes sure no three points lie on a single line. Find $\left\lvert y-x \right\rvert$.
[i]Proposed by Robin Park[/i]
2005 Today's Calculation Of Integral, 90
Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$
where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
2011 AMC 12/AHSME, 22
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\ 1500 \qquad
\textbf{(B)}\ 1560 \qquad
\textbf{(C)}\ 2320 \qquad
\textbf{(D)}\ 2480 \qquad
\textbf{(E)}\ 2500$
2019 LIMIT Category B, Problem 2
The digit in unit place of $1!+2!+\ldots+99!$ is
$\textbf{(A)}~3$
$\textbf{(B)}~0$
$\textbf{(C)}~1$
$\textbf{(D)}~7$
2013 AMC 12/AHSME, 15
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 $
2001 IMO Shortlist, 2
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.
2003 Austrian-Polish Competition, 7
Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $.
Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .
2023 Singapore Senior Math Olympiad, 4
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
Mexican Quarantine Mathematical Olympiad, #5
Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$:
$$p \mid f(n)f(p-1)!+n^{f(p)}.$$
[i]Proposed by Dorlir Ahmeti[/i]
2025 Bundeswettbewerb Mathematik, 2
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
1969 IMO Shortlist, 64
$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$
2024 AMC 12/AHSME, 2
What is $10! - 7! \cdot 6!$?
$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
2015 Tuymaada Olympiad, 4
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]
KoMaL A Problems 2019/2020, A. 770
Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.
2008 Indonesia TST, 2
Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.
2000 AIME Problems, 14
Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m),$ meaning that \[ k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m, \] where each $f_i$ is an integer, $0\le f_i\le i,$ and $0<f_m.$ Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\cdots+1968!-1984!+2000!,$ find the value of $f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j.$
2007 Harvard-MIT Mathematics Tournament, 1
Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
2020 AMC 8 -, 12
For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation?
$$5! \cdot 9! = 12 \cdot N!$$
$\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$
2013 Dutch IMO TST, 1
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.
2010 China Team Selection Test, 2
Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.
1999 Poland - Second Round, 6
Suppose that $a_1,a_2,...,a_n$ are integers such that $a_1 +2^ia_2 +3^ia_3 +...+n^ia_n = 0$ for $i = 1,2,...,k -1$, where $k \ge 2$ is a given integer. Prove that $a_1+2^ka_2+3^ka_3+...+n^ka_n$ is divisible by $k!$.