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, 45
Tags: induction , divisibility theory
Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$. Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$.

PEN A Problems, 79
Tags: divisibility theory
Determine all pairs of integers $(a, b)$ such that \[\frac{a^{2}}{2ab^{2}-b^{3}+1}\] is a positive integer.

PEN A Problems, 81
Tags: divisibility theory
Determine all triples of positive integers $(a, m, n)$ such that $a^m +1$ divides $(a+1)^n$.

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]

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, 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, 69
Tags: modular arithmetic , number theory , prime factorization , divisibility theory
Prove that if the odd prime $p$ divides $a^{b}-1$, where $a$ and $b$ are positive integers, then $p$ appears to the same power in the prime factorization of $b(a^{d}-1)$, where $d=\gcd(b,p-1)$.

PEN A Problems, 23
Tags: quadratic , gauss , modular arithmetic , number theory , relatively prime , divisibility theory
(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

PEN A Problems, 4
Tags: search , polynomial , vieta , symmetry , quadratic , divisibility theory , algebra
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.

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, 13
Tags: floor function , inequalities , divisibility theory , logarithm
Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

PEN A Problems, 105
Tags: number theory , least common multiple , divisibility theory
Find the smallest positive integer $n$ such that [list][*] $n$ has exactly $144$ distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of $n$. [/list]

PEN A Problems, 72
Tags: modular arithmetic , number theory , divisibility theory
Determine all pairs $(n,p)$ of nonnegative integers such that [list] [*] $p$ is a prime, [*] $n<2p$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

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, 52
Tags: quadratic , modular arithmetic , divisibility theory
Let $d$ be any positive integer not equal to 2, 5, or 13. Show that one can find distinct $a$ and $b$ in the set $\{2,5,13,d\}$ such that $ab - 1$ is not a perfect square.

PEN A Problems, 37
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.

1998 Romania Team Selection Test, 3
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, 11
Tags: pigeonhole principle , linear algebra , matrix , modular arithmetic , divisibility theory
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, 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, 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, 111
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, 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 $d_{9}-d_{8}=17$.

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

PEN A Problems, 28
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 $n \ge m \ge 1$.