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Found problems: 3

2024 Stars of Mathematics, P2
Tags: double factorial , number theory
For any positive integer $n$ we define $n!!=\prod_{k=0}^{\lceil n/2\rceil -1}(n-2k)$. Prove that if the positive integers $a,b,c$ satisfy $a!=b!!+c!!$, then $b$ and $c$ are odd. [i]Proposed by Mihai Cipu[/i]

1946 Moscow Mathematical Olympiad, 114
Tags: double factorial , factorial
Prove that for any positive integer $n$ the following identity holds $\frac{(2n)!}{n!}= 2^n \cdot (2n - 1)!!$

2009 Mathcenter Contest, 1
Tags: number theory , factorial , double factorial
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]