Olympiad problems

2022 CIIM, 6

Prove that $\tau ((n+1)!) \leq 2 \tau (n!)$ for all positive integers $n$.

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?

1969 AMC 12/AHSME, 23

Tags: factorial , number theory , prime number

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

2012 BMT Spring, 2

Tags: factorial , number theory , prime factorization

Find the smallest number with exactly 28 divisors.

2005 Greece JBMO TST, 4

Tags: number theory , divide , factorial

Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$

2003 India Regional Mathematical Olympiad, 7

Tags: factorial , ratio

Consider the set $X$ = $\{ 1,2 \ldots 10 \}$ . Find two disjoint nonempty sunsets $A$ and $B$ of $X$ such that a) $A \cup B = X$; b) $\prod_{x\in A}x$ is divisible by $\prod_{x\in B}x$, where $\prod_{x\in C}x$ is the product of all numbers in $C$; c) $\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}$ is as small as possible.

2011 Math Prize for Girls Olympiad, 4

Tags: linear algebra , matrix , factorial , inequalities , function , geometric series

Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.

2022 SG Originals, Q3

Tags: functional equation , number theory , factorial , function

Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

1953 Moscow Mathematical Olympiad, 256

Tags: equation , factorial , algebra

Find roots of the equation $$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$

1940 Moscow Mathematical Olympiad, 067

Tags: compare , algebra , factorial

Which is greater: $300!$ or $100^{300}$?

2022/2023 Tournament of Towns, P1

Tags: number theory , factorial

Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.

2019 Simon Marais Mathematical Competition, B2

Tags: factorial , number theory

For each odd prime number $p$, prove that the integer $$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$ (Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)

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 $(n + 1)!$

2021 Macedonian Mathematical Olympiad, Problem 1

Tags: sequence , recursive , inequality , factorial , am-gm , inequalities

Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.

2009 Putnam, B1

Tags: factorial , induction , abstract algebra , vector , strong induction , logarithm

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$

2006 Thailand Mathematical Olympiad, 10

Tags: number theory , factorial , remainder

Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

2006 Cezar Ivănescu, 1

Tags: factorial , harmonic series , newton s binom , limit , integral , calculus

[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $ [b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $

2012 Centers of Excellency of Suceava, 3

Tags: factorial , series , definite integral , real analysis , integral sequence

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

2013 Dutch IMO TST, 1

Tags: factorial , number theory

Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

2019 Girls in Mathematics Tournament, 4

Tags: factorial , number theory , combinatorics

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?

1993 India National Olympiad, 5

Tags: factorial , floor function , search , combinatorics

Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.

2011 Saudi Arabia Pre-TST, 4.2

Tags: number theory , factorial

Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$

2016 AMC 12/AHSME, 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 AMC 8 -, 17

Tags: factorial

If $n$ is an even positive integer, the [i]double factorial[/i] notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum? $$2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!$$ $\textbf{(A)} ~0\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~8\qquad$

1978 Chisinau City MO, 159

Tags: number theory , prime , divide , divisible , factorial

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.