Olympiad problems

2015 Rioplatense Mathematical Olympiad, Level 3, 3

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.

2009 AIME Problems, 7

Tags: floor function , factorial , function

Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$.

2007 ITest, 6

Tags: factorial

Find the units digit of the sum \[(1!)^2+(2!)^2+(3!)^2+(4!)^2+\cdots+(2007!)^2.\] $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }5\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }9$

2009 Putnam, B1

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

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!}.$

2005 Today's Calculation Of Integral, 90

Tags: calculus , limit , logarithm , integration , function , factorial , ratio

Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$ where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

2018 Thailand TSTST, 1

Tags: factorial , number theory

Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$ where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$.

1969 Canada National Olympiad, 6

Tags: induction , factorial

Find the sum of $1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!$, where $n!=n(n-1)(n-2)\cdots2\cdot1$.

1994 Canada National Olympiad, 1

Tags: algebra , factorial , polynomial

Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .

1967 IMO Longlists, 2

Tags: factorial , inequality

Prove that \[\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}},\] and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$

2005 IMO Shortlist, 4

Tags: number theory , modular arithmetic , divisibility , exponential , chinese remainder theorem , factorial

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

1997 VJIMC, Problem 4-M

Tags: real analysis , trigonometry , limit , summation , factorial

Prove that $$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$

1969 IMO Longlists, 64

Tags: inequality , factorial , algebra

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

2012-2013 SDML (High School), 3

Tags: factorial

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

2022 Belarusian National Olympiad, 9.2

Tags: algebra , factorial , inequalities

Prove the inequality $$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.

2000 AIME Problems, 7

Tags: floor function , factorial

Given that \[ \frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!} \] find the greatest integer that is less than $\frac N{100}.$

2006 Cezar Ivănescu, 1

Tags: limit , factorial , harmonic series , newton s binom , 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 $

1970 IMO Longlists, 24

Tags: factorial , inequalities

Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds.

2012 ELMO Shortlist, 4

Tags: number theory , factorial

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]

1978 Chisinau City MO, 159

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

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.

2004 Alexandru Myller, 4

Tags: limit , analysis , real analysis , definite integral , factorial , function

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

2006 JBMO ShortLists, 6

Tags: number theory , floor function , factorial

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$.

2007 Iran MO (3rd Round), 8

Tags: limit , floor function , logarithm , factorial , trigonometry , ceiling function

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]

2007 IMO Shortlist, 7

Tags: number theory , modular arithmetic , 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]

2019 Taiwan TST Round 1, 2

Tags: number theory , factorial

Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.

2013 NIMO Problems, 3

Tags: floor function , factorial

Let $a_1, a_2, \dots, a_{1000}$ be positive integers whose sum is $S$. If $a_n!$ divides $n$ for each $n = 1, 2, \dots, 1000$, compute the maximum possible value of $S$. [i]Proposed by Michael Ren[/i]