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

2018 Balkan MO Shortlist, N2
Tags: factorial , number theory
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

1969 IMO Shortlist, 15
Tags: inequalities , factorial , algebra , combinatorics , combinatorial inequality
$(CZS 4)$ 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 $K = K_1 + \cdots + K_n$

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 $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.

1996 Tournament Of Towns, (496) 3
Tags: factorial , number theory , perfect square
Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square? (S Tokarev)

Mexican Quarantine Mathematical Olympiad, #5
Tags: factorial , number theory , functional equation
Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$: $$p \mid f(n)f(p-1)!+n^{f(p)}.$$ [i]Proposed by Dorlir Ahmeti[/i]

2020 Estonia Team Selection Test, 1
Tags: number theory , least common multiple , factorial
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

2011 Dutch Mathematical Olympiad, 1
Tags: number theory , factorial , diophantine , diophantine equation
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

1969 IMO Shortlist, 64
Tags: inequality , factorial , algebra
$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

2019 AMC 10, 14
Tags: factorial
The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$? $\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17 $

2012-2013 SDML (High School), 3
Tags: factorial
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

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! $

1934 Eotvos Mathematical Competition, 1
Tags: number theory , integer , factorial
Let $n$ be a given positive integer and $$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$ Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.

2004 Alexandru Myller, 4
Tags: analysis , real analysis , definite integral , factorial , function , limit
For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

1997 VJIMC, Problem 4-M
Tags: trigonometry , summation , real analysis , factorial , limit
Prove that $$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$

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

2009 Mathcenter Contest, 1
Tags: double factorial , factorial , number theory
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]

2008 Tournament Of Towns, 4
Tags: number theory , factorial , divisible
Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

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 $

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$

KoMaL A Problems 2019/2020, A. 770
Tags: number theory , fibonacci , factorial
Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

PEN F Problems, 13
Tags: factorial , rational number
Prove that numbers of the form \[\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)$ are rational if and only if starting from some $i$ on all the $a_{i}$'s are either equal to $0$ ( in which case the sum is finite) or all are equal to $i-1$.

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

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

2007 Harvard-MIT Mathematics Tournament, 1
Tags: factorial , floor function
Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

2021 Azerbaijan Senior NMO, 1
Tags: number theory , factorial , prime factorization
At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?