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, 106
Tags: floor function , divisibility theory
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.

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 $n$.

PEN A Problems, 39
Tags: divisibility theory
Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]

PEN A Problems, 20
Tags: induction , abstract algebra , modular arithmetic , quadratic , 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, 104
Tags: divisibility theory
A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.

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, 9
Tags: number theory , relatively prime , divisibility theory
Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

PEN A Problems, 15
Tags: number theory , least common multiple , divisibility theory
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, 55
Tags: divisibility theory
Show that for every natural number $n$ the product \[\left( 4-\frac{2}{1}\right) \left( 4-\frac{2}{2}\right) \left( 4-\frac{2}{3}\right) \cdots \left( 4-\frac{2}{n}\right)\] is an integer.

PEN A Problems, 107
Tags: inequalities , rearrangement inequality , divisibility theory , logarithm
Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.

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, 114
Tags: number theory , greatest common divisor , modular arithmetic , divisibility theory
What is the greatest common divisor of the set of numbers \[\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?\]

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, 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, 67
Tags: divisibility theory
Prove that $2n \choose n$ is divisible by $n+1$.

PEN A Problems, 65
Tags: divisibility theory
Clara computed the product of the first $n$ positive integers and Valerid computed the product of the first $m$ even positive integers, where $m \ge 2$. They got the same answer. Prove that one of them had made a mistake.

PEN A Problems, 102
Tags: divisibility theory
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$

1993 Iran MO (2nd round), 1
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, 24
Tags: floor function , modular arithmetic , divisibility theory
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, 2
Tags: arithmetic sequence , diophantine equation , divisibility theory , pell s equation
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.

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$.