This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 310

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

2023 SG Originals, Q4
Tags: factorial , number theory
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.

2023 Bulgarian Spring Mathematical Competition, 12.3
Tags: factorial , number theory
Given is a polynomial $f$ of degree $m$ with integer coefficients and positive leading coefficient. A positive integer $n$ is $\textit {good for f(x)}$ if there exists a positive integer $k_n$, such that $n!+1=f(n)^{k_n}$. Prove that there exist only finitely many integers good for $f$.

1986 AMC 12/AHSME, 10
Tags: algorithm , factorial
The 120 permutations of the AHSME are arranged in dictionary order as if each were an ordinary five-letter word. The last letter of the 85th word in this list is: $ \textbf{(A)}\ \text{A} \qquad \textbf{(B)}\ \text{H} \qquad \textbf{(C)}\ \text{S} \qquad \textbf{(D)}\ \text{M} \qquad \textbf{(E)}\ \text{E} $

2005 Today's Calculation Of Integral, 79
Tags: calculus , integration , geometry , algebra , function , domain , factorial
Find the area of the domain expressed by the following system inequalities. \[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]

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.

2001 Denmark MO - Mohr Contest, 2
Tags: number theory , factorial , digit , last digit
If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end? (With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).

2004 District Olympiad, 2
Tags: factorial , algebra , equalities
Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.

1967 IMO Shortlist, 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 $n = 1.$

1972 IMO Longlists, 15
Tags: factorial , number theory , recurrence relation , binomial coefficient
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.