Olympiad problems

PEN A Problems, 84

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]

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, 40

Tags: greatest common divisor , divisibility theory , number theory

Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

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 $n \ge m \ge 1$.

PEN A Problems, 90

Tags: divisibility theory

Determine all pairs $(x, y)$ of positive integers with $y \vert x^2 +1$ and $x^2 \vert y^3 +1$.

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, 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

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, 61

Tags: search , divisibility theory

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, 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 \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]

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]

PEN A Problems, 113

Tags: number theory , relatively prime , divisibility theory

Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]

PEN A Problems, 7

Tags: quadratic , modular arithmetic , divisibility theory

Let $n$ be a positive integer such that $2+2\sqrt{28n^2 +1}$ is an integer. Show that $2+2\sqrt{28n^2 +1}$ is the square of an integer.

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, 26

Tags: floor function , inequalities , induction , ratio , divisibility theory

Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.

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, 89

Tags: divisibility theory

Determine all pairs $(a, b)$ of integers for which $a^{2}+b^{2}+3$ is divisible by $ab$.

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, 108

Tags: arithmetic sequence , divisibility theory

For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]

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, 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, 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, 83

Tags: floor function , divisibility theory

Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.

PEN A Problems, 47

Tags: factorial , floor function , number theory , prime number , divisibility theory

Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.

PEN A Problems, 44

Tags: algebra , polynomial , complex number , divisibility theory

Suppose that $4^{n}+2^{n}+1$ is prime for some positive integer $n$. Show that $n$ must be a power of $3$.