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

2008 Indonesia TST, 2
Tags: factorial , positive integer , number theory
Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.

2019 IMO, 4
Tags: number theory , factorial
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \] [i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]

1965 AMC 12/AHSME, 33
Tags: factorial
If the number $ 15!$, that is, $ 15 \cdot 14 \cdot 13 \dots 1$, ends with $ k$ zeros when given to the base $ 12$ and ends with $ h$ zeros when given to the base $ 10$, then $ k \plus{} h$ equals: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2009 Federal Competition For Advanced Students, P1, 1
Tags: inequalities , factorial , exponential , exponential inequality , diophantine , induction , algebra
Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

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

2012 BMT Spring, 2
Tags: prime factorization , factorial , number theory
Find the smallest number with exactly 28 divisors.

2021 European Mathematical Cup, 3
Tags: number theory , factorial , perfect square
Let $\ell$ be a positive integer. We say that a positive integer $k$ is [i]nice [/i] if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers. \\ \\ (Théo Lenoir)

2004 Thailand Mathematical Olympiad, 9
Tags: sum , factorial , algebra
Compute the sum $$\sum_{k=0}^{n}\frac{(2n)!}{k!^2(n-k)!^2}.$$

1986 Austrian-Polish Competition, 7
Tags: factorial , divide , prime divisor
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 $n!$.

1940 Moscow Mathematical Olympiad, 067
Tags: compare , factorial , algebra
Which is greater: $300!$ or $100^{300}$?

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

1993 India National Olympiad, 5
Tags: search , floor function , combinatorics , factorial
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.

2006 QEDMO 2nd, 8
Tags: number theory , factorial , perfect cube
Show that for any positive integer $n\ge 4$, there exists a multiple of $n^3$ between $n!$ and $(n + 1)!$

2002 Manhattan Mathematical Olympiad, 3
Tags: factorial
The product $1 \cdot 2 \cdot \ldots \cdot n$ is denoted by $n!$ and called [i]n-factorial[/i]. Prove that the product \[ 1!2!3!\ldots 49!51! \ldots 100! \] (the factor $50!$ is missing) is the square of an integer number.

2015 Tuymaada Olympiad, 4
Tags: algebra , factorial
Let $n!=ab^2$ where $a$ is free from squares. Prove, that for every $\epsilon>0$ for every big enough $n$ it is true, that $$2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}$$ [i]M. Ivanov[/i]

2002 AMC 10, 22
Tags: floor function , factorial
In how many zeroes does the number $\dfrac{2002!}{(1001!)^2}$ end? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }200\qquad\textbf{(E) }400$

2017 Singapore Junior Math Olympiad, 2
Tags: factorial , diophantine , diophantine equation , number theory
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

2013 Dutch IMO TST, 1
Tags: number theory , factorial
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

2000 Singapore Senior Math Olympiad, 2
Tags: number theory , factorial , diophantine equation , diophantine
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.

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]

2014 NIMO Problems, 5
Tags: function , floor function , factorial
Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]

2002 Germany Team Selection Test, 3
Tags: decimal representation , number theory , factorial , digit
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 $k$.

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

1986 Tournament Of Towns, (129) 4
Tags: number theory , divisible , factorial , 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)

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]