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

PEN A Problems, 82
Tags: quadratics , 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, 66
Tags: Divisibility Theory
(Four Number Theorem) Let $a, b, c,$ and $d$ be positive integers such that $ab=cd$. Show that there exists positive integers $p, q, r,s$ such that \[a=pq, \;\; b=rs, \;\; c=ps, \;\; d=qr.\]

PEN A Problems, 52
Tags: quadratics , modular arithmetic , Divisibility Theory , pen
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, 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, 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, 49
Tags: inequalities , Divisibility Theory
Prove that there is no positive integer $n$ such that, for $k=1, 2, \cdots, 9,$ the leftmost digit of $(n+k)!$ equals $k$.

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

PEN A Problems, 16
Tags: induction , Divisibility Theory
Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.

1993 Iran MO (2nd round), 1
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 $43$.

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, 73
Tags: number theory , Divisibility Theory , pen
Determine all pairs $(n,p)$ of positive integers such that [list][*] $p$ is a prime, $n>1$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

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, 42
Tags: modular arithmetic , Divisibility Theory , pen
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, 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, 57
Tags: Divisibility Theory
Prove that for every $n \in \mathbb{N}$ the following proposition holds: $7|3^n +n^3$ if and only if $7|3^{n} n^3 +1$.

PEN A Problems, 80
Tags: inequalities , algebra , polynomial , Divisibility Theory
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, 31
Tags: Diophantine equation , Divisibility Theory , pen
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.

PEN A Problems, 17
Tags: modular arithmetic , quadratics , number theory , relatively prime , Divisibility Theory , pen
Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.

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, 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, 46
Tags: Divisibility Theory
Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.