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

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

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 $n+a_n$.

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

2017 Regional Olympiad of Mexico Southeast, 4
Tags: factorial , diophantine equation , number theory
Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

2006 Putnam, A2
Tags: polynomial , function , probability , factorial , logarithm , algebra
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

2024 Poland - Second Round, 6
Tags: number theory , factorial
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 $$\prod_{i=1}^{p}(p^i)!.$$

2015 Finnish National High School Mathematics Comp, 3
Tags: number theory , factorial , factor , exponential , divide
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

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 $n\geq 0.$

1986 AMC 12/AHSME, 10
Tags: factorial , algorithm
The 120 permutations of the AHSME are arranged in dictionary order as if each were an ordinary five-letter word. The last letter of the 85th word in this list is: $ \textbf{(A)}\ \text{A} \qquad \textbf{(B)}\ \text{H} \qquad \textbf{(C)}\ \text{S} \qquad \textbf{(D)}\ \text{M} \qquad \textbf{(E)}\ \text{E} $

2006 Putnam, B6
Tags: integration , limit , factorial , college , inequalities
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define \[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\] for $n\ge 0.$ Evaluate \[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

2016 Belarus Team Selection Test, 3
Tags: number theory , factorial , divisibility
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

1984 Brazil National Olympiad, 1
Tags: number theory , factorial , positive integer , factorial equation , diophantine equation
Find all solutions in positive integers to $(n+1)^k -1 = n!$

2006 Thailand Mathematical Olympiad, 10
Tags: number theory , factorial , remainder
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

2014 Argentine National Olympiad, Level 3, 5.
Tags: number theory , factorial
An integer $n \geq 3$ is called [i]special[/i] if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$. Find all special numbers $n$ such that $10 \leq n \leq 100$.

2007 Purple Comet Problems, 12
Tags: counting , distinguishability , factorial
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.

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

2022 SG Originals, Q3
Tags: function , functional equation , factorial , number theory
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

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

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

2009 Mathcenter Contest, 1
Tags: number theory , factorial , double factorial
For any natural $n$ , define $n!!=(n!)!$ e.g. $3!!=(3!)!=6!=720$. Let $a_1,a_2,...,a_n$ be a positive integer Prove that $$\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!}$$ is an integer. [i](nooonuii)[/i]

2011 Saudi Arabia BMO TST, 4
Tags: algebra , factorial
Consider a non-zero real number $a$ such that $\{a\} + \left\{\frac{1}{a}\right\}=1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.

2018 Abels Math Contest (Norwegian MO) Final, 1
Tags: factorial , modulo , residue , number theory
For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$. How many different residues modulo $1000$ do you get from $n!!$ for $n= 1, 3, 5, …$?

2008 Moldova Team Selection Test, 4
Tags: calculus , polynomial , factorial , algebra , integration
A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

2024 Bangladesh Mathematical Olympiad, P9
Tags: number theory , factorial , divisibility , p-adic
Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.

2000 Harvard-MIT Mathematics Tournament, 34
Tags: factorial , hmmt
What is the largest $n$ such that $n! + 1$ is a square?