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, 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, 110
Tags: divisibility theory , relatively prime , number theory
For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.

PEN A Problems, 44
Tags: complex number , divisibility theory , algebra , polynomial
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, 61
Tags: divisibility theory , search
For any positive integer $n>1$, let $p(n)$ be the greatest prime divisor of $n$. Prove that there are infinitely many positive integers $n$ with \[p(n)<p(n+1)<p(n+2).\]

PEN A Problems, 95
Tags: divisibility theory
Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$?

2018 Moldova Team Selection Test, 12
Tags: divisibility theory , modular arithmetic , floor function
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, 13
Tags: logarithm , divisibility theory , inequalities , floor function
Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

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, 40
Tags: number theory , divisibility theory , greatest common divisor
Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

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

1993 Iran MO (2nd round), 1
Tags: divisibility theory , modular arithmetic
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, 94
Tags: algorithm , divisibility theory
Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.

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.

2010 Postal Coaching, 3
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, 49
Tags: divisibility theory , inequalities
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, 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 $a^2 +b^2 +c^2 +1 = d^2$.

PEN A Problems, 16
Tags: divisibility theory , induction
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, 35
Tags: divisibility theory , modular arithmetic
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 $p^2$.

1998 Romania Team Selection Test, 3
Tags: inequalities , divisibility theory , induction
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

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, 85
Tags: divisibility theory
Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.

PEN A Problems, 15
Tags: least common multiple , divisibility theory , number 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, 14
Tags: divisibility theory , number theory , modular arithmetic
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.

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