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.

AND:
OR:
NO:

Found problems: 121

2010 Postal Coaching, 3
Tags: divisibility theory
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.

PEN A Problems, 10
Tags: euler , number theory , divisibility theory
Let $n$ be a positive integer with $n \ge 3$. Show that \[n^{n^{n^{n}}}-n^{n^{n}}\] is divisible by $1989$.

PEN A Problems, 63
Tags: induction , pigeonhole principle , divisibility theory
There is a large pile of cards. On each card one of the numbers $1$, $2$, $\cdots$, $n$ is written. It is known that the sum of all numbers of all the cards is equal to $k \cdot n!$ for some integer $k$. Prove that it is possible to arrange cards into $k$ stacks so that the sum of numbers written on the cards in each stack is equal to $n!$.

PEN A Problems, 31
Tags: diophantine equation , divisibility theory
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.

PEN A Problems, 104
Tags: divisibility theory
A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.

2002 Bosnia Herzegovina Team Selection Test, 3
Tags: number theory , greatest common divisor , modular arithmetic , relatively prime , divisibility theory
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

PEN A Problems, 101
Tags: divisibility theory
Find all composite numbers $n$ having the property that each proper divisor $d$ of $n$ has $n-20 \le d \le n-12$.

PEN A Problems, 115
Tags: divisibility theory
Does there exist a $4$-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of $1992$?

PEN A Problems, 35
Tags: modular arithmetic , divisibility theory
Let $p \ge 5$ be a prime number. Prove that there exists an integer $a$ with $1 \le a \le p-2$ such that neither $a^{p-1} -1$ nor $(a+1)^{p-1} -1$ is divisible by $p^2$.

PEN A Problems, 62
Tags: divisibility theory
Let $p(n)$ be the greatest odd divisor of $n$. Prove that \[\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.\]

1998 Romania Team Selection Test, 3
Tags: induction , divisibility theory , inequalities
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

PEN A Problems, 75
Tags: modular arithmetic , divisibility theory
Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.

PEN A Problems, 96
Tags: integration , modular arithmetic , number theory , relatively prime , divisibility theory
Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

1989 India National Olympiad, 4
Tags: floor function , divisibility theory
Determine all $n \in \mathbb{N}$ for which [list][*] $n$ is not the square of any integer, [*] $\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$. [/list]

PEN A Problems, 110
Tags: number theory , relatively prime , divisibility theory
For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.

PEN A Problems, 15
Tags: number theory , least common multiple , divisibility theory
Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.

PEN A Problems, 67
Tags: divisibility theory
Prove that $2n \choose n$ is divisible by $n+1$.

PEN A Problems, 85
Tags: divisibility theory
Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.

PEN A Problems, 24
Tags: floor function , modular arithmetic , divisibility theory
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

PEN A Problems, 107
Tags: inequalities , rearrangement inequality , divisibility theory , logarithm
Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.

PEN A Problems, 20
Tags: induction , abstract algebra , modular arithmetic , quadratic , number theory , divisibility theory
Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.