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, 110
Tags: number theory , relatively prime , divisibility 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, 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, 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, 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, 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, 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, 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, 68
Tags: least common multiple , greatest common divisor , divisibility theory , number 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, 94
Tags: algorithm , divisibility theory
Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.

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, 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, 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, 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, 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, 70
Tags: binomial coefficient , induction , divisibility theory
Suppose that $m=nq$, where $n$ and $q$ are positive integers. Prove that the sum of binomial coefficients \[\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}\] is divisible by $m$.

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, 76
Tags: divisibility theory
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[(a-1)(b-1)(c-1)\hspace{0.2in}\text{is a divisor of}\hspace{0.2in}abc-1.\]

PEN A Problems, 75
Tags: modular arithmetic , divisibility theory
Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.

PEN A Problems, 33
Tags: divisibility theory
Let $a,b,x\in \mathbb{N}$ with $b>1$ and such that $b^{n}-1$ divides $a$. Show that in base $b$, the number $a$ has at least $n$ non-zero digits.

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

PEN A Problems, 19
Tags: induction , modular arithmetic , divisibility theory , number 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, 56
Tags: modular arithmetic , induction , divisibility theory
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, 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 $p$?

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, 93
Tags: divisibility theory
Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.