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

2023 Bangladesh Mathematical Olympiad, P1
Tags: diophantine equation , factorial , number theory
Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$ Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

2022 Belarusian National Olympiad, 9.2
Tags: algebra , inequalities , factorial
Prove the inequality $$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.

2006 JBMO ShortLists, 6
Tags: factorial , floor function , number theory
Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n\minus{}1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor$.

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

2019 Latvia Baltic Way TST, 16
Tags: algebra , system of equations , factorial
Determine all tuples of positive integers $(x, y, z, t)$ such that: $$ xyz = t!$$ $$ (x+1)(y+1)(z+1) = (t+1)!$$ holds simultaneously.

2011 QEDMO 9th, 4
Tags: factorial , number theory , multiple , divide
Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

2007 IMO Shortlist, 7
Tags: modular arithmetic , number theory , prime factorization , factorial , combinatorial 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]

2014-2015 SDML (Middle School), 5
Tags: factorial
In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

2017 USA TSTST, 4
Tags: factorial , number theory , diophantine equation
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$. [i]Proposed by Mark Sellke[/i]

2005 iTest, 37
Tags: number theory , factorial
How many zeroes appear at the end of $209$ factorial?

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.

2010 AIME Problems, 3
Tags: factorial , number theory
Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

2008 Stanford Mathematics Tournament, 9
Tags: factorial
What is the sum of the prime factors of 20!?

2006 Putnam, B6
Tags: limit , factorial , college , inequalities , integration
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define \[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\] for $n\ge 0.$ Evaluate \[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

2003 AMC 12-AHSME, 23
Tags: factorial
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$

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

2015 Kazakhstan National Olympiad, 5
Tags: factorial
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 $n \ge 4$.

2008 Indonesia TST, 2
Tags: number theory , factorial , positive integer
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.

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]

1994 Canada National Olympiad, 1
Tags: factorial , polynomial , algebra
Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .

2005 AIME Problems, 5
Tags: counting , distinguishability , factorial
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

1999 AMC 12/AHSME, 25
Tags: factorial
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$

2017 Singapore Junior Math Olympiad, 2
Tags: diophantine , diophantine equation , number theory , factorial
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 $a_1,a_2,...,a_{2n}$

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 $k!=1\cdot2\cdot\ldots\cdot k$.

2015 District Olympiad, 1
Tags: inequalities , induction , factorial
For any $ n\ge 2 $ natural, show that the following inequality holds: $$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$