Olympiad problems

2019 Serbia JBMO TST, 1

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

2024 Philippine Math Olympiad, P2

Tags: factorial , number theory

Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that \[\dfrac{(2^n+1)!!-1}{2^{n+1}}\] is an integer.

2023 Grand Duchy of Lithuania, 4

Tags: number theory , divide , factorial

Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$ Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.

2015 Finnish National High School Mathematics Comp, 3

Tags: number theory , factorial , factor , exponential , divide

Determine the largest integer $k$ for which $12^k$ is a factor of $120! $

2015 Purple Comet Problems, 14

Tags: factorial

Find the greatest positive integer $n$ so that $3^n$ divides $70! + 71! + 72!.$

2000 China Team Selection Test, 2

Tags: induction , algebra , polynomial , factorial , combinatorics , finite differences

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example $\frac{10}{9}=\frac{2!5!}{3!3!3!}$.

1964 All Russian Mathematical Olympiad, 048

Tags: number theory , factorial

Find all the natural $n$ such that $n!$ is not divisible by $n^2$.

2008 Tournament Of Towns, 4

Tags: number theory , factorial , divisible

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

2004 Mediterranean Mathematics Olympiad, 1

Tags: factorial , inequalities , prime factorization , number theory

Find all natural numbers $m$ such that \[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]