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

2024 AMC 12/AHSME, 2
Tags: factorial
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 $

1978 AMC 12/AHSME, 21
Tags: factorial , number theory
$p$ 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.

2017 IFYM, Sozopol, 2
Tags: factorial , number theory
Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example $\frac{10}{9}=\frac{2!5!}{3!3!3!}$.

2005 AIME Problems, 5
Tags: distinguishability , counting , factorial
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

2008 Tournament Of Towns, 4
Tags: divisible , number theory , factorial
Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

2017 Saudi Arabia BMO TST, 1
Tags: divide , factorial , number theory , divisible
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

2011 Puerto Rico Team Selection Test, 6
Tags: factorial
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: )

2006 AIME Problems, 4
Tags: factorial , number theory , prime factorization
Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000.

1964 All Russian Mathematical Olympiad, 048
Tags: number theory , factorial
Find all the natural $n$ such that $n!$ is not divisible by $n^2$.

2019 Purple Comet Problems, 17
Tags: factorial , number theory
Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.

Mexican Quarantine Mathematical Olympiad, #5
Tags: factorial , functional equation , number theory
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]

1996 Tournament Of Towns, (496) 3
Tags: factorial , perfect square , number theory
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)

2016 SGMO, Q5
Tags: factorial , modular arithmetic , number theory
Let $d_{m} (n)$ denote the last non-zero digit of $n$ in base $m$ where $m,n$ are naturals. Given distinct odd primes $p_1,p_2,\ldots,p_k$, show that there exists infinitely many natural $n$ such that $$d_{2p_i} (n!) \equiv 1 \pmod {p_i}$$ for all $i = 1,2,\ldots,k$.

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

2016 AMC 12/AHSME, 1
Tags: fraction , factorial
What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

2001 Denmark MO - Mohr Contest, 2
Tags: factorial , digit , last digit , number theory
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 $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).

2020 Switzerland - Final Round, 5
Tags: number theory , diophantine , diophantine equation , factorial
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

2013 Portugal MO, 4
Tags: factorial , divisor , number theory
Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?

2006 MOP Homework, 4
Tags: factorial , number theory
Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.

2021 Miklós Schweitzer, 2
Tags: factorial , diophantine equation , number theory
Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

1992 ITAMO, 3
Tags: number theory , divisor , sum of divisors , factorial
Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

1934 Eotvos Mathematical Competition, 1
Tags: number theory , integer , factorial
Let $n$ be a given positive integer and $$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$ Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.

2014 Online Math Open Problems, 25
Tags: induction , combinatorial geometry , factorial
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]

2022 Singapore MO Open, Q3
Tags: factorial , function , number theory , functional equation
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

1999 Harvard-MIT Mathematics Tournament, 7
Tags: factorial
Evaluate $\sum_{n=1}^\infty \dfrac{n^5}{n!}.$