Found problems: 310
2009 Tournament Of Towns, 4
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$
[i](8 points)[/i]
2017 Peru IMO TST, 4
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
1999 AMC 12/AHSME, 25
There are unique integers $ a_2, a_3, a_4, a_5, a_6, a_7$ such that \[ \frac {5}{7} \equal{} \frac {a_2}{2!} \plus{} \frac {a_3}{3!} \plus{} \frac {a_4}{4!} \plus{} \frac {a_5}{5!} \plus{} \frac {a_6}{6!} \plus{} \frac {a_7}{7!},\] where $ 0 \le a_i < i$ for $ i \equal{} 2,3...,7$. Find $ a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 \plus{} a_6 \plus{} a_7$.
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ 11 \qquad
\textbf{(E)}\ 12$
2025 Nepal National Olympiad, 4
Find all pairs of positive integers \( n \) and \( x \) such that
\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]
[i](Petko Lazarov, Bulgaria)[/i]
1996 AMC 12/AHSME, 3
$\frac{(3!)!}{3!} =$
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 6\qquad \text{(D)}\ 40\qquad \text{(E)}\ 120$
2016 Brazil Team Selection Test, 2
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
2020 AMC 12/AHSME, 6
For all integers $n \geq 9,$ the value of
$$\frac{(n+2)!-(n+1)!}{n!}$$
is always which of the following?
$\textbf{(A) } \text{a multiple of }4 \qquad \textbf{(B) } \text{a multiple of }10 \qquad \textbf{(C) } \text{a prime number} \\ \textbf{(D) } \text{a perfect square} \qquad \textbf{(E) } \text{a perfect cube}$
2003 AMC 12-AHSME, 23
How many perfect squares are divisors of the product $ 1!\cdot 2!\cdot 3!\cdots 9!$?
$ \textbf{(A)}\ 504 \qquad \textbf{(B)}\ 672 \qquad \textbf{(C)}\ 864 \qquad \textbf{(D)}\ 936 \qquad \textbf{(E)}\ 1008$
2018 Romania Team Selection Tests, 4
Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.
2013 AMC 12/AHSME, 15
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2019 Malaysia National Olympiad, 3
A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?
1966 IMO Shortlist, 11
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 $x \le y$ ?
KoMaL A Problems 2019/2020, A. 770
Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.
2019 Latvia Baltic Way TST, 16
Determine all tuples of positive integers $(x, y, z, t)$ such that:
$$ xyz = t!$$
$$ (x+1)(y+1)(z+1) = (t+1)!$$
holds simultaneously.
2019 Girls in Mathematics Tournament, 4
A positive integer $n$ is called [i]cute[/i] when there is a positive integer $m$ such that $m!$ ends in exactly $n$ zeros.
a) Determine if $2019$ is cute.
b) How many positive integers less than $2019$ are cute?
2017 JBMO Shortlist, NT2
Determine all positive integers n such that $n^2/ (n - 1)!$
2012 NIMO Problems, 3
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]
1970 IMO Longlists, 3
Prove that $(a!\cdot b!) | (a+b)!$ $\forall a,b\in\mathbb{N}$.
2023 SG Originals, Q4
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
2024 AMC 10, 2
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
$
2015 Kazakhstan National Olympiad, 5
Find all possible $\{ x_1,x_2,...x_n \}$ permutations of $ \{1,2,...,n \}$ so that when $1\le i \le n-2 $ then we have $x_i < x_{i+2}$ and when $1 \le i \le n-3$ then we have $x_i < x_{i+3}$ . Here $n \ge 4$.
2025 Bundeswettbewerb Mathematik, 2
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
2000 AIME Problems, 14
Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m),$ meaning that \[ k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m, \] where each $f_i$ is an integer, $0\le f_i\le i,$ and $0<f_m.$ Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\cdots+1968!-1984!+2000!,$ find the value of $f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j.$
2000 Belarusian National Olympiad, 2
Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$
2009 Princeton University Math Competition, 2
Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$.