Found problems: 310
Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$. Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$. Determine all conditions under which equality holds.
Show that
$$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$
for every integer $n\geq 0.$
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.
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 $.
Prove that the equation
\[ 2^x + 5^y - 31^z = n! \]
has only a finite number of non-negative integer solutions $x,y,z,n$.
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
Which of the following numbers is a perfect square?
$ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $
Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.
What is the value of $\dfrac{11!-10!}{9!}$?
$\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$
Show that
\[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\]
is an integer, for any positive integer \(n\).
[b](a)[/b] Prove that for all positive integers $m,n$ we have
\[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\]
[b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that
\[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\]
Has a limit.
Prove the inequality
$$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.
For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$.
[i]Author: Tejaswi Navilarekkallu, India[/i]
Find the integer $n$ such that $6!\times7!=n!$.
[i]2017 CCA Math Bonanza Individual Round #1[/i]
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}
$
Find all the natural $n$ such that $n!$ is not divisible by $n^2$.
Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.
If $n$ is an even positive integer, the [i]double factorial[/i] notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum? $$2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!$$
$\textbf{(A)} ~0\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~8\qquad$
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.
Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$
Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$
Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
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
$
Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$
Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.