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

2015 District Olympiad, 1
Tags: inequalities , induction , factorial
For any $ n\ge 2 $ natural, show that the following inequality holds: $$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$

2019 IMO Shortlist, N1
Tags: number theory , factorial
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \] [i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]

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

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? $\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$

2011-2012 SDML (High School), 12
Tags: factorial
Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate? $\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$

2011 QEDMO 9th, 4
Tags: multiple , divide , factorial , number theory
Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

2013 Turkey MO (2nd round), 1
Tags: number theory , diophantine equation , factorial
Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

1988 Iran MO (2nd round), 1
Tags: algebra , factorial , induction , polynomial
[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

2005 Today's Calculation Of Integral, 79
Tags: geometry , integration , algebra , function , factorial , domain , calculus
Find the area of the domain expressed by the following system inequalities. \[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]

2008 Harvard-MIT Mathematics Tournament, 13
Tags: algebra , factorial , polynomial
Let $ P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that \[ P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0. \]Determine the value of $ P(2008)$. You may use factorials in your answer.

2004 District Olympiad, 2
Tags: algebra , factorial , equalities
Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.

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.

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

2007 Romania Team Selection Test, 1
Tags: floor function , function , factorial , inequalities , number theory
Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

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.

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

2016 India PRMO, 2
Tags: diophantine , algebra , factorial , equation , diophantine equation , wilson
Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$

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

2021 Irish Math Olympiad, 1
Tags: factorial , number theory
Let $N = 15! = 15\cdot 14\cdot 13 ... 3\cdot 2\cdot 1$. Prove that $N$ can be written as a product of nine different integers all between $16$ and $30$ inclusive.

2024 Belarusian National Olympiad, 9.7
Tags: factorial , number theory
Find all pairs of positive integers $(m,n)$, for which $$(m^n-n)^m=n!+m$$ [i]D. Volkovets[/i]

2019 Simon Marais Mathematical Competition, B2
Tags: number theory , factorial
For each odd prime number $p$, prove that the integer $$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$ (Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)

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} $

2008 National Olympiad First Round, 26
Tags: factorial
Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$. What is the remainder when $11!\cdot 12! \cdot A$ is divided by $11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10 $

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

2022 AMC 8 -, 17
Tags: factorial
If $n$ is an even positive integer, the [i]double factorial[/i] notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum? $$2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!$$ $\textbf{(A)} ~0\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~8\qquad$