Olympiad problems

2015 AMC 10, 23

Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? $ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $

2025 Nepal National Olympiad, 4

Tags: number theory , factorial

Find all pairs of positive integers $ n $ and $ x $ such that \[ 1^n + 2^n + 3^n + \cdots + n^n = x! \] [i](Petko Lazarov, Bulgaria)[/i]

2000 Harvard-MIT Mathematics Tournament, 34

Tags: factorial , hmmt

What is the largest $n$ such that $n! + 1$ is a square?

2007 Romania Team Selection Test, 1

Tags: function , inequalities , floor function , factorial , number theory

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

2024 Olimphíada, 1

Tags: number theory , least common multiple , factorial

Find all pairs of positive integers $(m,n)$ such that $$lcm(1,2,\dots,n)=m!$$ where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.

PEN F Problems, 13

Tags: rational number , factorial

Prove that numbers of the form \[\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)$ are rational if and only if starting from some $i$ on all the $a_{i}$'s are either equal to $0$ ( in which case the sum is finite) or all are equal to $i-1$.

2024 Mongolian Mathematical Olympiad, 1

Tags: number theory , diophantine equation , factorial

Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\] [i]Proposed by Otgonbayar Uuye.[/i]

2019 Purple Comet Problems, 17

Tags: factorial , number theory

Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.

2006 Germany Team Selection Test, 2

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

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]

1949-56 Chisinau City MO, 7

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

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

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 $.

1999 Harvard-MIT Mathematics Tournament, 7

Tags: factorial

Evaluate $\sum_{n=1}^\infty \dfrac{n^5}{n!}.$

2016 Belarus Team Selection Test, 3

Tags: factorial , number theory , divisibility

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

2013 Princeton University Math Competition, 4

Tags: factorial

Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers). Note: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.

2008 AIME Problems, 6

Tags: induction , ratio , factorial

The sequence $ \{a_n\}$ is defined by \[ a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2. \]The sequence $ \{b_n\}$ is defined by \[ b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2. \]Find $ \frac {b_{32}}{a_{32}}$.

2023 SG Originals, Q4

Tags: factorial , number theory

Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.

1940 Moscow Mathematical Olympiad, 056

Tags: decimal representation , number theory , factorial

How many zeros does $100!$ have at its end in the usual decimal representation?

2008 Pre-Preparation Course Examination, 5

Tags: probability , factorial , expected value , probability and stats

A permutation $ \pi$ is selected randomly through all $ n$-permutations. a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$ b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$

2000 Harvard-MIT Mathematics Tournament, 26

Tags: factorial

What are the last $3$ digits of $1!+2!+\cdots +100!$

2018 Malaysia National Olympiad, B3

Tags: number theory , factorial

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

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 $n = 1.$

2016 India PRMO, 2

Tags: algebra , factorial , equation , diophantine , diophantine equation , wilson

Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$

2009 Mathcenter Contest, 1

Tags: double factorial , factorial , number theory

For any natural $n$ , define $n!!=(n!)!$ e.g. $3!!=(3!)!=6!=720$. Let $a_1,a_2,...,a_n$ be a positive integer Prove that $$\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!}$$ is an integer. [i](nooonuii)[/i]

1965 AMC 12/AHSME, 33

Tags: factorial

If the number $ 15!$, that is, $ 15 \cdot 14 \cdot 13 \dots 1$, ends with $ k$ zeros when given to the base $ 12$ and ends with $ h$ zeros when given to the base $ 10$, then $ k \plus{} h$ equals: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2024 Israel TST, P1

Tags: factorial , number theory , floor function

For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\] Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.