This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 310

2014 JHMMC 7 Contest, 9
Tags: factorial
Let $n!=n\cdot (n-1)\cdot (n-2)\cdot \ldots \cdot 2\cdot 1$.For example, $5! = 5\cdot 4\cdot 3 \cdot 2\cdot 1 = 120.$ Compute $\frac{(6!)^2}{5!\cdot 7!}$.

2003 Austrian-Polish Competition, 7
Tags: number theory , divide , factorial , divisible
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 $(n^n)!$ .

2005 IMO Shortlist, 4
Tags: factorial , modular arithmetic , number theory , divisibility , exponential , chinese remainder theorem
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

2016 IFYM, Sozopol, 5
Tags: number theory , factorial
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.

2019 International Zhautykov OIympiad, 1
Tags: combinatorics , factorial
Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

1974 Putnam, B5
Tags: factorial , inequalities
Show that $$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$ for every integer $n\geq 0.$

1972 IMO, 3
Tags: factorial , number theory , recurrence relation , binomial coefficient
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

1961 AMC 12/AHSME, 35
Tags: factorial
The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ . . . a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k$, and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$ ${{ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ 4} $

2024 Turkey Team Selection Test, 4
Tags: number theory , modular arithmetic , olympiad number theory , factorial , perfect square
Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.

2013 AMC 12/AHSME, 15
Tags: factorial , induction , number theory , prime factorization
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 $

1968 All Soviet Union Mathematical Olympiad, 102
Tags: number theory , factorial
Prove that you can represent an arbitrary number not exceeding $n!$ as a sum of $k$ different numbers ($k\le n$) that are divisors of $n!$.

1998 Estonia National Olympiad, 4
Tags: factorial , diophantine , diophantine equation , number theory
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

2015 AMC 10, 23
Tags: factorial , modular arithmetic , floor function
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? $ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $

2012-2013 SDML (Middle School), 3
Tags: factorial
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

2012 ELMO Shortlist, 4
Tags: factorial , number theory
Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

2022 Grosman Mathematical Olympiad, P1
Tags: factorial , number theory
For each positive integer $n$ denote: \[n!=1\cdot 2\cdot 3\dots n\] Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

2006 Thailand Mathematical Olympiad, 10
Tags: number theory , factorial , remainder
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

1990 AIME Problems, 11
Tags: factorial
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.

2017 USA TSTST, 4
Tags: factorial , number theory , diophantine equation
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$. [i]Proposed by Mark Sellke[/i]

2014 Middle European Mathematical Olympiad, 4
Tags: factorial , algorithm , relatively prime , number theory
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]

1969 IMO Longlists, 64
Tags: inequality , factorial , algebra
$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

2000 Singapore Senior Math Olympiad, 2
Tags: factorial , diophantine equation , diophantine , number theory
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.

2023 Singapore Senior Math Olympiad, 4
Tags: factorial , number theory
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.

2016 Abels Math Contest (Norwegian MO) Final, 2b
Tags: factorial , diophantine equation , diophantine , number theory
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

2004 Thailand Mathematical Olympiad, 9
Tags: algebra , sum , factorial
Compute the sum $$\sum_{k=0}^{n}\frac{(2n)!}{k!^2(n-k)!^2}.$$