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 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 $$\prod_{i=1}^{p}(p^i)!.$$

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? $\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$

2020 AMC 8 -, 12
Tags: factorial
For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation? $$5! \cdot 9! = 12 \cdot N!$$ $\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$

2012 NIMO Problems, 3
Tags: factorial
Let \[ S = \sum_{i = 1}^{2012} i!. \] The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$. [i]Proposed by Lewis Chen[/i]

2012 AMC 12/AHSME, 11
Tags: probability , counting , distinguishability , factorial
Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round? $ \textbf{(A)}\ \frac{5}{72}\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{1}{6}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ 1 $

2022 Irish Math Olympiad, 1
Tags: number theory , factorial
1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i]. Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.

1966 IMO Longlists, 11
Tags: number theory , factorial , additive number theory
Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?

1965 AMC 12/AHSME, 33
Tags: factorial
If the number $ 15!$, that is, $ 15 \cdot 14 \cdot 13 \dots 1$, ends with $ k$ zeros when given to the base $ 12$ and ends with $ h$ zeros when given to the base $ 10$, then $ k \plus{} h$ equals: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2015 Middle European Mathematical Olympiad, 7
Tags: factorial , diophantine equation , number theory , exponential equation
Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

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

1993 India National Olympiad, 5
Tags: factorial , floor function , search , combinatorics
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.

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 \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

2008 Putnam, B2
Tags: integration , limit , calculus , factorial , logarithm
Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

2019 Simon Marais Mathematical Competition, B2
Tags: factorial , number theory
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} $

2015 Tuymaada Olympiad, 4
Tags: factorial , algebra
Let $n!=ab^2$ where $a$ is free from squares. Prove, that for every $\epsilon>0$ for every big enough $n$ it is true, that $$2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}$$ [i]M. Ivanov[/i]

2005 Greece JBMO TST, 4
Tags: number theory , divide , factorial
Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$

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]

2020 LIMIT Category 1, 19
Tags: limit , algebra , factorial
Let $a=2019^{1009}, b=2019!$ and $c=1010^{2019}$, then which of the following is true? (A)$c<b<a$ (B)$a<b<c$ (C)$b<a<c$ (D)$b<c<a$

2003 BAMO, 1
Tags: factorial , perfect number , number theory , divisor
An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$. Let $n!$ denote the product $1\cdot 2\cdot 3\cdot ...\cdot n$, where $n$ is a positive integer. An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$. For example, $24$ is a factorial number since $24 = 4! = 1\cdot 2\cdot 3\cdot 4$. Find all perfect numbers greater than $1$ that are also factorials.

2010 National Olympiad First Round, 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} $

2002 Germany Team Selection Test, 3
Tags: number theory , decimal representation , digit , factorial
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 $k$.

2010 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , factorial , integration
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.

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

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: )