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

2017 Singapore Senior Math Olympiad, 1
Tags: diophantine , diophantine equation , factorial , number theory
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

2006 Stanford Mathematics Tournament, 14
Tags: floor function , factorial , function , number theory , modular arithmetic
Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.

2022/2023 Tournament of Towns, P1
Tags: number theory , factorial
Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.

2013 Dutch IMO TST, 1
Tags: factorial , number theory
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

1989 Tournament Of Towns, (238) 2
Tags: combinatorics , factorial , subset
Consider all the possible subsets of the set $\{1,2,..., N\}$ which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is $(N + 1)! - 1$. (Based on idea of R.P. Stanley)

2020 China Girls Math Olympiad, 4
Tags: number theory , greatest common divisor , factorial
Let $p,q$ be primes, where $p>q$. Define $t=\gcd(p!-1,q!-1)$. Prove that $t\le p^{\frac{p}{3}}$.

2015 CCA Math Bonanza, I2
Tags: factorial
The operation $*$ is defined by the following: $a*b=a!-ab-b.$ Compute the value of $5*8.$ [i]2015 CCA Math Bonanza Individual Round #2[/i]

2012 Today's Calculation Of Integral, 803
Tags: calculus , integration , limit , factorial , calculus computations , logarithm
Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2008 Stanford Mathematics Tournament, 9
Tags: factorial
What is the sum of the prime factors of 20!?

2003 India Regional Mathematical Olympiad, 7
Tags: factorial , ratio
Consider the set $X$ = $\{ 1,2 \ldots 10 \}$ . Find two disjoint nonempty sunsets $A$ and $B$ of $X$ such that a) $A \cup B = X$; b) $\prod_{x\in A}x$ is divisible by $\prod_{x\in B}x$, where $\prod_{x\in C}x$ is the product of all numbers in $C$; c) $\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}$ is as small as possible.

2011 Saudi Arabia IMO TST, 1
Tags: number theory , remainder , factorial
Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.

2018-2019 SDML (High School), 1
Tags: factorial
Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.

2014 Singapore Senior Math Olympiad, 16
Tags: factorial
Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$

2023 Romania Team Selection Test, P3
Tags: number theory , factorial , divisibility
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]

2024 AMC 12/AHSME, 4
Tags: factorial
What is the least value of $n$ such that $n!$ is a multiple of $2024$? $ \textbf{(A) }11 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22 \qquad \textbf{(D) }23 \qquad \textbf{(E) }253 \qquad $

2017 Peru IMO TST, 4
Tags: factorial , number theory , perfect square
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?

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

2011 Math Prize for Girls Olympiad, 4
Tags: linear algebra , matrix , factorial , inequalities , function , geometric series
Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.

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

2023 Bangladesh Mathematical Olympiad, P1
Tags: factorial , diophantine equation , number theory
Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

2002 Germany Team Selection Test, 3
Tags: decimal representation , digit , factorial , number theory
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2015 IMO Shortlist, N2
Tags: factorial , divisibility , number theory
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

1970 IMO Longlists, 3
Tags: factorial
Prove that $(a!\cdot b!) | (a+b)!$ $\forall a,b\in\mathbb{N}$.

2014-2015 SDML (Middle School), 5
Tags: factorial
In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

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 $k + n$.