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

2010 China Team Selection Test, 2
Tags: factorial , prime number , number theory
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 $n+a_n$.

2010 AIME Problems, 3
Tags: factorial , number theory
Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

2008 Indonesia TST, 2
Tags: positive integer , factorial , number theory
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.

2004 Federal Competition For Advanced Students, Part 1, 3
Tags: factorial
For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$. [b](a)[/b] Prove that $Z(a, b)$ is an integer for $a \leq b$. [b](b)[/b] Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

2012 Abels Math Contest (Norwegian MO) Final, 3a
Tags: product , last digit , factorial , number theory
Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.

2024 Mongolian Mathematical Olympiad, 1
Tags: diophantine equation , number theory , factorial
Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] [i]Proposed by Otgonbayar Uuye.[/i]

2015 AMC 10, 23
Tags: modular arithmetic , floor function , factorial
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 $

2001 IMO, 4
Tags: global , factorial , modular arithmetic , combinatorics
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 $S(a)-S(b)$.

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

2019 CCA Math Bonanza, T4
Tags: factorial
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]

2021 Azerbaijan Senior NMO, 1
Tags: prime factorization , number theory , factorial
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?

2023 Romania Team Selection Test, P3
Tags: number theory , divisibility , factorial
Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$ [i]Proposed by Andrei Bâra[/i]

2018 Belarusian National Olympiad, 9.2
Tags: inequalities , factorial , number theory
For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$ where $k!=1\cdot2\cdot\ldots\cdot k$.

2016 APMC, 5
Tags: factorial , number theory
Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$ Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$

1991 IMO Shortlist, 15
Tags: periodic sequence , digit , decimal representation , factorial , modular arithmetic
Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

2006 AMC 10, 11
Tags: factorial
What is the tens digit in the sum $ 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!$? $ \textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$

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 $n!$.

1969 AMC 12/AHSME, 23
Tags: prime number , number theory , factorial
For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$ $\textbf{(D) }n-1\qquad \textbf{(E) }n$

1972 IMO Shortlist, 8
Tags: recurrence relation , binomial coefficient , factorial , number theory
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

2003 AIME Problems, 1
Tags: factorial
Given that \[ \frac{((3!)!)!}{3!} = k \cdot n!, \] where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n$.

2011 Saudi Arabia Pre-TST, 4.2
Tags: number theory , factorial
Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$

2024 Philippine Math Olympiad, P2
Tags: factorial , number theory
Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that \[\dfrac{(2^n+1)!!-1}{2^{n+1}}\] is an integer.

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

2020 Estonia Team Selection Test, 1
Tags: least common multiple , factorial , number theory
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

2000 China Team Selection Test, 2
Tags: finite differences , induction , factorial , combinatorics , polynomial , algebra
Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]