Found problems: 310
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$.
Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$
An integer $n \geq 3$ is called [i]special[/i] if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$. Find all special numbers $n$ such that $10 \leq n \leq 100$.
Find all natural numbers $m$ such that
\[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]
Answer the following questions:
(1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$
(2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$
An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself.
For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$.
Let $n!$ denote the product $1\cdot 2\cdot 3\cdot ...\cdot n$, where $n$ is a positive integer.
An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$.
For example, $24$ is a factorial number since $24 = 4! = 1\cdot 2\cdot 3\cdot 4$.
Find all perfect numbers greater than $1$ that are also factorials.
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .
Find all pairs of positive integers $(a,b)$ such that
$$a!+b!=a^b + b^a.$$
Let
\[
S = \sum_{i = 1}^{2012} i!.
\]
The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$.
[i]Proposed by Lewis Chen[/i]
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$.
Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for $n\ge 0.$ Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]
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$
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.
If the number $ 15!$, that is, $ 15 \cdot 14 \cdot 13 \dots 1$, ends with $ k$ zeros when given to the base $ 12$ and ends with $ h$ zeros when given to the base $ 10$, then $ k \plus{} h$ equals:
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
Evaluate $\sum_{n=1}^\infty \dfrac{n^5}{n!}.$
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$
Let $n$ be a given positive integer and
$$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$
Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.
Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.
Determine all tuples of positive integers $(x, y, z, t)$ such that:
$$ xyz = t!$$
$$ (x+1)(y+1)(z+1) = (t+1)!$$
holds simultaneously.
What is the largest 2-digit prime factor of the integer $n = \binom{200}{100}$?