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: 121

PEN A Problems, 83
Tags: divisibility theory , floor function
Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.

PEN A Problems, 42
Tags: divisibility theory , modular arithmetic
Suppose that $2^n +1$ is an odd prime for some positive integer $n$. Show that $n$ must be a power of $2$.

PEN A Problems, 54
Tags: divisibility theory
A natural number $n$ is said to have the property $P$, if whenever $n$ divides $a^{n}-1$ for some integer $a$, $n^2$ also necessarily divides $a^{n}-1$. [list=a] [*] Show that every prime number $n$ has the property $P$. [*] Show that there are infinitely many composite numbers $n$ that possess the property $P$. [/list]

PEN A Problems, 24
Tags: divisibility theory , floor function , modular arithmetic
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, 87
Tags: modular arithmetic , divisibility theory
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.

PEN A Problems, 6
Tags: vieta , divisibility theory , polynomial , algebra
[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

PEN A Problems, 113
Tags: divisibility theory , number theory , relatively prime
Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]

PEN A Problems, 100
Tags: integration , divisibility theory , modular arithmetic , calculus
Find all positive integers $n$ such that $n$ has exactly $6$ positive divisors $1<d_{1}<d_{2}<d_{3}<d_{4}<n$ and $1+n=5(d_{1}+d_{2}+d_{3}+d_{4})$.

PEN A Problems, 10
Tags: euler , divisibility theory , number 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, 118
Tags: divisibility theory , function
Determine the highest power of $1980$ which divides \[\frac{(1980n)!}{(n!)^{1980}}.\]

PEN A Problems, 68
Tags: divisibility theory , least common multiple , greatest common divisor , number theory
Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]

PEN A Problems, 71
Tags: function , divisibility theory
Determine all integers $n > 1$ such that \[\frac{2^{n}+1}{n^{2}}\] is an integer.

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

PEN A Problems, 80
Tags: inequalities , divisibility theory , polynomial , algebra
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

PEN A Problems, 27
Tags: divisibility theory , modular arithmetic , binomial coefficient
Show that the coefficients of a binomial expansion $(a+b)^n$ where $n$ is a positive integer, are all odd, if and only if $n$ is of the form $2^{k}-1$ for some positive integer $k$.

PEN A Problems, 34
Tags: divisibility theory
Let $p_{1}, p_{2}, \cdots, p_{n}$ be distinct primes greater than $3$. Show that \[2^{p_{1}p_{2}\cdots p_{n}}+1\] has at least $4^{n}$ divisors.

PEN A Problems, 8
Tags: quadratic , divisibility theory , search , modular arithmetic , algebra
The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

PEN A Problems, 92
Tags: divisibility theory
Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.

PEN A Problems, 78
Tags: divisibility theory
Determine all ordered pairs $(m, n)$ of positive integers such that \[\frac{n^{3}+1}{mn-1}\] is an integer.

PEN A Problems, 74
Tags: divisibility theory
Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.

PEN A Problems, 29
Tags: divisibility theory
For which positive integers $k$, is it true that there are infinitely many pairs of positive integers $(m, n)$ such that \[\frac{(m+n-k)!}{m! \; n!}\] is an integer?

PEN A Problems, 51
Tags: divisibility theory
Let $a,b,c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^{k}$ and $b+c=2^{m}$ for some integers $k$ and $m$, then $a=1$.

PEN A Problems, 91
Tags: divisibility theory
Determine all pairs $(a, b)$ of positive integers such that $ab^2+b+7$ divides $a^2 b+a+b$.

PEN A Problems, 30
Tags: divisibility theory
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.

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