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

2003 SNSB Admission, 3
Tags: prime number , ring theory , factorial , number theory
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.

1993 Bundeswettbewerb Mathematik, 4
Tags: factorial , number theory
Does there exist a non-negative integer n, such that the first four digits of n! is 1993?

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 \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]

1969 AMC 12/AHSME, 23
Tags: factorial , prime number , number theory
For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$ $\textbf{(D) }n-1\qquad \textbf{(E) }n$

2013 Turkey MO (2nd round), 1
Tags: diophantine equation , factorial , number theory
Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

2001 IMO Shortlist, 2
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 $S(a)-S(b)$.

2015 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: factorial , number theory
We say an integer number $n \ge 1$ is conservative, if the smallest prime divisor of $(n!)^n+1$ is at most $n+2015$. Decide if the number of conservative numbers is infinite or not.

2007 Iran MO (3rd Round), 8
Tags: trigonometry , limit , ceiling function , factorial , floor function , logarithm
In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]

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.

2006 Stanford Mathematics Tournament, 5
Tags: factorial , ratio , geometric series
A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term?

2010 Junior Balkan Team Selection Tests - Romania, 3
Tags: factorial , remainder , number theory
Determine the integers $n, n \ge 2$, with the property that the numbers $1! , 2 ! , 3 ! , ..., (n- 1)!$ give different remainders when dividing by $n $.

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

2004 Federal Competition For Advanced Students, Part 1, 3
Tags: factorial
For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$. [b](a)[/b] Prove that $Z(a, b)$ is an integer for $a \leq b$. [b](b)[/b] Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

2007 IMO Shortlist, 7
Tags: modular arithmetic , prime factorization , factorial , combinatorial number theory , number theory
For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$. [i]Author: Tejaswi Navilarekkallu, India[/i]

2007-2008 SDML (Middle School), 6
Tags: factorial
Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes.

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

2013 India Regional Mathematical Olympiad, 6
Tags: number theory , polynomial , root , factorial , greatest common divisor
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

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

2024 Philippine Math Olympiad, P2
Tags: factorial , number theory
Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that \[\dfrac{(2^n+1)!!-1}{2^{n+1}}\] is an integer.

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.

2014 NIMO Problems, 5
Tags: floor function , 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]

1940 Moscow Mathematical Olympiad, 062
Tags: 3-digit , factorial , number theory
Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.

1957 Moscow Mathematical Olympiad, 369
Tags: combinatorics , factorial , minimum , min
Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.

2018 Belarusian National Olympiad, 9.2
Tags: number theory , inequalities , factorial
For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$ where $k!=1\cdot2\cdot\ldots\cdot k$.

2019 Purple Comet Problems, 17
Tags: factorial , number theory
Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.