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, 11
Tags: pigeonhole principle , linear algebra , matrix , modular arithmetic , Divisibility Theory , pen
Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

PEN A Problems, 50
Tags: Divisibility Theory
Show that every integer $k>1$ has a multiple less than $k^4$ whose decimal expansion has at most four distinct digits.

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, 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, 93
Tags: Divisibility Theory
Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.

PEN A Problems, 6
Tags: algebra , polynomial , Vieta , Divisibility Theory , pen
[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, 87
Tags: modular arithmetic , Divisibility Theory
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.

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, 88
Tags: induction , function , modular arithmetic , Divisibility Theory
Find all positive integers $n$ such that $9^{n}-1$ is divisible by $7^n$.

PEN A Problems, 86
Tags: inequalities , induction , Divisibility Theory
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

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

PEN A Problems, 15
Tags: number theory , least common multiple , Divisibility Theory , pen
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, 109
Tags: Divisibility Theory
Find all positive integers $a$ and $b$ such that \[\frac{a^{2}+b}{b^{2}-a}\text{ and }\frac{b^{2}+a}{a^{2}-b}\] are both integers.

PEN A Problems, 77
Tags: Divisibility Theory
Find all positive integers, representable uniquely as \[\frac{x^{2}+y}{xy+1},\] where $x$ and $y$ are positive integers.

PEN A Problems, 21
Tags: Divisibility Theory
Let n be a positive integer. Show that the product of $ n$ consecutive positive integers is divisible by $ n!$

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

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]

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, 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, 44
Tags: algebra , polynomial , complex numbers , 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, 64
Tags: quadratics , Divisibility Theory
The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.

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

PEN A Problems, 14
Tags: Putnam , 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, 32
Tags: floor function , Divisibility Theory
Let $ a$ and $ b$ be natural numbers such that \[ \frac{a}{b}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}. \] Prove that $ a$ is divisible by $ 1979$.

PEN A Problems, 68
Tags: number theory , least common multiple , greatest common divisor , Divisibility 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 \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]