Olympiad problems

2019 LIMIT Category A, Problem 7

The digit in unit place of $1!+2!+\ldots+99!$ is $\textbf{(A)}~3$ $\textbf{(B)}~0$ $\textbf{(C)}~1$ $\textbf{(D)}~7$

1993 Bundeswettbewerb Mathematik, 4

Tags: number theory , factorial

Does there exist a non-negative integer n, such that the first four digits of n! is 1993?

2023 Belarus - Iran Friendly Competition, 1

Tags: factorial , number theory

Find all positive integers n such that the product $1! \cdot 2! \cdot \cdot \cdot \cdot n!$ is a perfect square

2010 National Olympiad First Round, 4

Tags: factorial

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

2016 Abels Math Contest (Norwegian MO) Final, 2b

Tags: diophantine equation , diophantine , factorial , number theory

Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

2006 Stanford Mathematics Tournament, 5

Tags: factorial , geometric series , ratio

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?

2019 Serbia JBMO TST, 1

Tags: number theory , divisible , factorial

Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?

1992 AIME Problems, 15

Tags: factorial , geometry , floor function

Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?

2022 JHMT HS, 2

Tags: factorial , number theory

Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.

2021 Macedonian Mathematical Olympiad, Problem 1

Tags: sequence , inequality , am-gm , recursive , factorial , 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 $\sqrt[n]{a_n} < \frac {n+1}{2}$.

2002 Olympic Revenge, 7

Tags: factorial , number theory

Show that \[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\] is an integer, for any positive integer $n$.

1940 Moscow Mathematical Olympiad, 056

Tags: number theory , decimal representation , factorial

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

2005 AMC 10, 15

Tags: 3d geometry , modular arithmetic , number theory , factorial , geometry

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

2016 USAMO, 2

Tags: factorial

Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.

2022 JBMO Shortlist, N1

Tags: sum of integers , shortlist , number theory , factorial

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

2013 AMC 10, 20

Tags: prime factorization , induction , number theory , factorial

The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

2014 JHMMC 7 Contest, 9

Tags: factorial

Let $n!=n\cdot (n-1)\cdot (n-2)\cdot \ldots \cdot 2\cdot 1$.For example, $5! = 5\cdot 4\cdot 3 \cdot 2\cdot 1 = 120.$ Compute $\frac{(6!)^2}{5!\cdot 7!}$.

2019 AMC 12/AHSME, 4

Tags: factorial

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

PEN H Problems, 15

Tags: diophantine equation , number theory , factorial

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

2019 AMC 10, 25

Tags: factorial

For how many integers $n$ between $1$ and $50$, inclusive, is \[ \frac{(n^2-1)!}{(n!)^n} \]an integer? (Recall that $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

2020 LIMIT Category 1, 19

Tags: limit , factorial , algebra

Let $a=2019^{1009}, b=2019!$ and $c=1010^{2019}$, then which of the following is true? (A)$c<b<a$ (B)$a<b<c$ (C)$b<a<c$ (D)$b<c<a$

1972 IMO Longlists, 15

Tags: binomial coefficient , number theory , factorial , recurrence relation

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

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 USAMO, 4

Tags: induction , function , factorial , number theory , algebra , modular arithmetic

Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.

2024 AMC 12/AHSME, 4

Tags: factorial

What is the least value of $n$ such that $n!$ is a multiple of $2024$? $ \textbf{(A) }11 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22 \qquad \textbf{(D) }23 \qquad \textbf{(E) }253 \qquad $