Olympiad problems

1961 Putnam, B5

Tags: factorial , inequalities

Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define $$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$ Prove either part of the inequality $$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$ where the middle term has $k$ factorial symbols.

2017 Regional Olympiad of Mexico Southeast, 4

Tags: number theory , diophantine equation , factorial

Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

2005 AMC 10, 22

Tags: factorial

For how many positive integers $ n$ less than or equal to $ 24$ is $ n!$ evenly divisible by $ 1 \plus{} 2 \plus{} \dots \plus{} n$? $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 21$

1982 Tournament Of Towns, (025) 3

Tags: number theory , diophantine , diophantine equation , factorial

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

2014 Online Math Open Problems, 25

Tags: induction , function , integration , calculus , factorial , derivative

If \[ \sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq \] for relatively prime positive integers $p,q$, find $p+q$. [i]Proposed by Michael Kural[/i]

2018 Rio de Janeiro Mathematical Olympiad, 1

Tags: factorial , rio

A natural number is a [i]factorion[/i] if it is the sum of the factorials of each of its decimal digits. For example, $145$ is a factorion because $145 = 1! + 4! + 5!$. Find every 3-digit number which is a factorion.

1940 Moscow Mathematical Olympiad, 062

Tags: 3-digit , number theory , factorial

Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.

2021 European Mathematical Cup, 3

Tags: factorial , perfect square , number theory

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)

2020 AMC 12/AHSME, 6

Tags: factorial

For all integers $n \geq 9,$ the value of $$\frac{(n+2)!-(n+1)!}{n!}$$ is always which of the following? $\textbf{(A) } \text{a multiple of }4 \qquad \textbf{(B) } \text{a multiple of }10 \qquad \textbf{(C) } \text{a prime number} \\ \textbf{(D) } \text{a perfect square} \qquad \textbf{(E) } \text{a perfect cube}$

1991 IMO Shortlist, 15

Tags: factorial , modular arithmetic , periodic sequence , decimal representation , digit

Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

2019 Ramnicean Hope, 1

Tags: limit , factorial , trigonometric limit , real analysis

Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]

2011 Dutch Mathematical Olympiad, 1

Tags: number theory , factorial , diophantine , diophantine equation

Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

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]

2010 China Team Selection Test, 2

Tags: prime number , factorial , number theory

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

2020 Switzerland - Final Round, 5

Tags: diophantine equation , factorial , number theory , diophantine

Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

1992 ITAMO, 3

Tags: factorial , sum of divisors , number theory , divisor

Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

2019 AMC 10, 6

Tags: factorial

A positive integer $n$ satisfies the equation $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$? $\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }15$

2012 Today's Calculation Of Integral, 803

Tags: calculus , integration , limit , factorial , calculus computations , logarithm

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2013 Turkey MO (2nd round), 1

Tags: number theory , diophantine equation , factorial

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

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 $3^{n^2} > (n!)^4$ .

2004 Thailand Mathematical Olympiad, 9

Tags: algebra , sum , factorial

Compute the sum $$\sum_{k=0}^{n}\frac{(2n)!}{k!^2(n-k)!^2}.$$

1961 AMC 12/AHSME, 35

Tags: factorial

The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ . . . a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k$, and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$ ${{ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ 4} $

2017 JBMO Shortlist, NT2

Tags: number theory , divisibility , factorial

Determine all positive integers n such that $n^2/ (n - 1)!$

2018 Balkan MO Shortlist, N2

Tags: factorial , number theory

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

2008 Moldova Team Selection Test, 4

Tags: polynomial , factorial , calculus , integration , algebra

A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.