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.

AND:
OR:
NO:

Found problems: 310

2024 AMC 10, 5
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 $

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 $x \le y$ ?

1986 Austrian-Polish Competition, 7
Tags: prime divisor , factorial , divide
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.

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.

2019 CCA Math Bonanza, T4
Tags: factorial
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\] [i]2019 CCA Math Bonanza Team Round #4[/i]

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 $k$.

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]

2005 IMO Shortlist, 4
Tags: factorial , modular arithmetic , number theory , divisibility , exponential , chinese remainder theorem
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

2006 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: factorial , induction , inequalities , number theory
(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$. (b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]

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

2016 SGMO, Q5
Tags: factorial , modular arithmetic , number theory
Let $d_{m} (n)$ denote the last non-zero digit of $n$ in base $m$ where $m,n$ are naturals. Given distinct odd primes $p_1,p_2,\ldots,p_k$, show that there exists infinitely many natural $n$ such that $$d_{2p_i} (n!) \equiv 1 \pmod {p_i}$$ for all $i = 1,2,\ldots,k$.

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

2016 USAMO, 2
Tags: factorial
Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.

2011 VTRMC, Problem 3
Tags: factorial , summation , real analysis
Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.

2013 Portugal MO, 4
Tags: number theory , factorial , divisor
Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?

2009 Math Prize For Girls Problems, 18
Tags: factorial
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$, where $ a$, $ b$, and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$?

1983 Canada National Olympiad, 1
Tags: factorial , number theory
Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$.

2008 iTest Tournament of Champions, 1
Tags: factorial
Find the remainder when $712!$ is divided by $719$.

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 $S(a)-S(b)$.

2017 CCA Math Bonanza, I1
Tags: factorial
Find the integer $n$ such that $6!\times7!=n!$. [i]2017 CCA Math Bonanza Individual Round #1[/i]

1992 Putnam, B2
Tags: factorial , college
For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^{k}$ in the expansion $(1+x+x^{2}+x^{3})^{n}$. Prove that $Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}$. [hide="hint"] Think of $\binom{n}{j}$ as the number of ways you can pick the $x^{2}$ term in the expansion.[/hide]

2007 Putnam, 3
Tags: binomial coefficient , probability , counting , distinguishability , factorial , function
Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.

2022 Grosman Mathematical Olympiad, P1
Tags: factorial , number theory
For each positive integer $n$ denote: \[n!=1\cdot 2\cdot 3\dots n\] Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

1972 IMO, 3
Tags: binomial coefficient , factorial , number theory , recurrence relation
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

1969 IMO Shortlist, 15
Tags: inequalities , factorial , combinatorics , combinatorial inequality , algebra
$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$