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

2002 Olympic Revenge, 7
Tags: factorial , number theory
Show that \[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\] is an integer, for any positive integer \(n\).

2012 ELMO Shortlist, 4
Tags: factorial , number theory
Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

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.

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 $

2019 International Zhautykov OIympiad, 1
Tags: combinatorics , factorial
Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

2014 AMC 10, 8
Tags: factorial
Which of the following numbers is a perfect square? $ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $

2016 Abels Math Contest (Norwegian MO) Final, 2b
Tags: factorial , diophantine equation , diophantine , number theory
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

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 $n= 1, 3, 5, …$?

2003 SNSB Admission, 3
Tags: number theory , prime number , ring theory , factorial
Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.

2014 JHMMC 7 Contest, 9
Tags: factorial
Let $n!=n\cdot (n-1)\cdot (n-2)\cdot \ldots \cdot 2\cdot 1$.For example, $5! = 5\cdot 4\cdot 3 \cdot 2\cdot 1 = 120.$ Compute $\frac{(6!)^2}{5!\cdot 7!}$.

2024 AMC 10, 2
Tags: factorial
What is $10! - 7! \cdot 6!$? $ \textbf{(A) }-120 \qquad \textbf{(B) }0 \qquad \textbf{(C) }120 \qquad \textbf{(D) }600 \qquad \textbf{(E) }720 \qquad $

1986 Tournament Of Towns, (129) 4
Tags: factorial , number theory , divisible , divisor , divide
We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

1939 Eotvos Mathematical Competition, 2
Tags: number theory , factorial , power of 2 , divide
Determine the highest power of $2$ that divides $2^n!$.

PEN H Problems, 15
Tags: diophantine equation , factorial , number theory
Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

1966 IMO Shortlist, 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 $x \le y$ ?

2020 Tournament Of Towns, 2
Tags: factorial , combinatorics , game , number theory
Three legendary knights are fighting against a multiheaded dragon. Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads? Alexey Zaslavsky

2000 Tuymaada Olympiad, 4
Tags: factorial , number theory
Prove that no number of the form $10^{-n}$, $n\geq 1,$ can be represented as the sum of reciprocals of factorials of different positive integers.

2016 AMC 10, 1
Tags: factorial , fraction
What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

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.

1968 All Soviet Union Mathematical Olympiad, 102
Tags: number theory , factorial
Prove that you can represent an arbitrary number not exceeding $n!$ as a sum of $k$ different numbers ($k\le n$) that are divisors of $n!$.

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 $(1, 2,..., n)> (n - 1) !$ .

1992 AIME Problems, 15
Tags: factorial , floor function , geometry
Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?

1970 IMO Longlists, 24
Tags: factorial , inequalities
Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds.

Kvant 2022, M2716
Tags: number theory , factorial
Find all pairs of natural numbers $(k, m)$ such that for any natural $n{}$ the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!{}$. [i]Proposed by P. Kozhevnikov[/i]

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)