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, 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 $7|3^{n} n^3 +1$.

2002 Bosnia Herzegovina Team Selection Test, 3
Tags: greatest common divisor , relatively prime , modular arithmetic , divisibility theory , number 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, 56
Tags: divisibility theory , modular arithmetic , induction
Let $a, b$, and $c$ be integers such that $a+b+c$ divides $a^2 +b^2 +c^2$. Prove that there are infinitely many positive integers $n$ such that $a+b+c$ divides $a^n +b^n +c^n$.

PEN A Problems, 82
Tags: function , quadratic , vieta , divisibility theory , vieta jumping , inequalities
Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

PEN A Problems, 37
Tags: greatest common divisor , relatively prime , divisibility theory , number theory , modular arithmetic
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

1989 India National Olympiad, 4
Tags: floor function , divisibility theory
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, 7
Tags: modular arithmetic , quadratic , 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.

PEN A Problems, 60
Tags: divisibility theory
Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number.

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, 69
Tags: modular arithmetic , prime factorization , divisibility theory , number 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, 97
Tags: divisibility theory
Suppose that $n$ is a positive integer and let \[d_{1}<d_{2}<d_{3}<d_{4}\] be the four smallest positive integer divisors of $n$. Find all integers $n$ such that \[n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.\]

PEN A Problems, 19
Tags: induction , modular arithmetic , number theory , divisibility theory
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.

PEN A Problems, 64
Tags: quadratic , divisibility theory
The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.

PEN A Problems, 59
Tags: divisibility theory
Suppose that $n$ has (at least) two essentially distinct representations as a sum of two squares. Specifically, let $n=s^{2}+t^{2}=u^{2}+v^{2}$, where $s \ge t \ge 0$, $u \ge v \ge 0$, and $s>u$. Show that $\gcd(su-tv, n)$ is a proper divisor of $n$.

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, 58
Tags: divisibility theory
Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge \tfrac{3k}{4}$. Let $n$ be a composite integer. Prove that [list=a] [*] if $n=2p_k$, then $n$ does not divide $(n-k)!$, [*] if $n>2p_k$, then $n$ divides $(n-k)!$. [/list]

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

PEN A Problems, 73
Tags: number theory , divisibility theory
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, 18
Tags: algebra , quadratic , difference of squares , special factorizations , divisibility theory
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

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

PEN A Problems, 28
Tags: function , algebra , floor function , number theory , greatest common divisor , 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, 48
Tags: divisibility theory
Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.

PEN A Problems, 98
Tags: divisibility theory
Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$, all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\]

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$.

PEN A Problems, 31
Tags: diophantine equation , divisibility theory
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.