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, 42
Tags: modular arithmetic , divisibility theory
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, 12
Tags: divisibility theory
Let $k,m,$ and $n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_{s}=s(s+1).$ Prove that the product \[(c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k})\] is divisible by the product $c_{1}c_{2}\cdots c_{n}$.

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, 14
Tags: modular arithmetic , number theory , divisibility theory
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.

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, 18
Tags: quadratic , algebra , difference of squares , special factorizations , divisibility theory
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

PEN A Problems, 44
Tags: algebra , polynomial , complex number , divisibility theory
Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.

PEN A Problems, 7
Tags: quadratic , modular arithmetic , divisibility theory
Let $n$ be a positive integer such that $2+2\sqrt{28n^2 +1}$ is an integer. Show that $2+2\sqrt{28n^2 +1}$ is the square of an integer.

PEN A Problems, 113
Tags: number theory , relatively prime , divisibility theory
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, 85
Tags: divisibility theory
Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.

PEN A Problems, 48
Tags: divisibility theory
Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.

PEN A Problems, 82
Tags: quadratic , function , inequalities , vieta , divisibility theory , vieta jumping
Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

PEN A Problems, 43
Tags: modular arithmetic , divisibility theory
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.

PEN A Problems, 59
Tags: divisibility theory
Suppose that $n$ has (at least) two essentially distinct representations as a sum of two squares. Specifically, let $n=s^{2}+t^{2}=u^{2}+v^{2}$, where $s \ge t \ge 0$, $u \ge v \ge 0$, and $s>u$. Show that $\gcd(su-tv, n)$ is a proper divisor of $n$.

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

PEN A Problems, 100
Tags: calculus , integration , modular arithmetic , divisibility theory
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, 41
Tags: modular arithmetic , divisibility theory
Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.

PEN A Problems, 116
Tags: divisibility theory
What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits $2$ through $9$?

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, 53
Tags: divisibility theory
Suppose that $x, y,$ and $z$ are positive integers with $xy=z^2 +1$. Prove that there exist integers $a, b, c,$ and $d$ such that $x=a^2 +b^2$, $y=c^2 +d^2$, and $z=ac+bd$.

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, 112
Tags: algebra , vieta , induction , number theory , relatively prime , divisibility theory
Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.

PEN A Problems, 89
Tags: divisibility theory
Determine all pairs $(a, b)$ of integers for which $a^{2}+b^{2}+3$ is divisible by $ab$.

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

2017 Pan-African Shortlist, N1
Tags: number theory , greatest common divisor , floor function , function , algebra , divisibility theory
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.