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

2005 AMC 10, 15
Tags: geometry , 3d geometry , factorial , modular arithmetic , number theory
How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

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 $n\geq 0.$

2018 AMC 10, 3
Tags: factorial
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire? $\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$

1983 AIME Problems, 8
Tags: factorial
What is the largest 2-digit prime factor of the integer $n = \binom{200}{100}$?

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

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 $k + n$.

2009 Tournament Of Towns, 4
Tags: factorial , function , floor function , number theory
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$ [i](8 points)[/i]

2022 JBMO Shortlist, N1
Tags: factorial , sum of integers , shortlist , number theory
Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

2018 Thailand TSTST, 1
Tags: number theory , factorial
Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$ where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$.

2018 CCA Math Bonanza, I1
Tags: factorial
What is the tens digit of the sum \[\left(1!\right)^2+\left(2!\right)^2+\left(3!\right)^2+\ldots+\left(2018!\right)^2?\] [i]2018 CCA Math Bonanza Individual Round #1[/i]

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 $t\le p^{\frac{p}{3}}$.

1983 Austrian-Polish Competition, 8
Tags: number theory , recurrence relation , sequence , divide , factorial , divisible , product
(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

2019 AMC 10, 2
Tags: factorial
What is the hundreds digit of $(20!-15!)?$ $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$

2010 Contests, 4
Tags: factorial
How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

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

2011 Saudi Arabia BMO TST, 1
Tags: product , algebra , factorial
Let $n$ be a positive integer. Find all real numbers $x_1,x_2 ,..., x_n$ such that $$\prod_{k=1}^{n}(x_k^2+ (k + 2)x_k + k^2 + k + 1) =\left(\frac{3}{4}\right)^n (n!)^2$$

2016 Taiwan TST Round 2, 1
Tags: factorial , number theory , divisibility
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

1989 Tournament Of Towns, (238) 2
Tags: factorial , subset , combinatorics
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)

2024 Poland - Second Round, 6
Tags: factorial , number theory
Given is a prime number $p$. Prove that the number $$p \cdot (p^2 \cdot \frac{p^{p-1}-1}{p-1})!$$ is divisible by $$\prod_{i=1}^{p}(p^i)!.$$

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

2019 Malaysia National Olympiad, 3
Tags: factorial , number theory
A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?

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 $7^3$.

PEN A Problems, 47
Tags: factorial , floor function , number theory , prime number , divisibility theory
Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.

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

2013 AMC 10, 20
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 $