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

PEN A Problems, 81
Tags: divisibility theory
Determine all triples of positive integers $(a, m, n)$ such that $a^m +1$ divides $(a+1)^n$.

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, 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, 84
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, 105
Tags: number theory , least common multiple , divisibility theory
Find the smallest positive integer $n$ such that [list][*] $n$ has exactly $144$ distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of $n$. [/list]

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 $\sqrt[3]{n}$.

PEN A Problems, 117
Tags: divisibility theory
Find the smallest positive integer $n$ such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers $m>1$.

PEN A Problems, 112
Tags: induction , algebra , relatively prime , vieta , divisibility theory , number theory
Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.

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

PEN A Problems, 45
Tags: divisibility theory , induction
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, 72
Tags: divisibility theory , modular arithmetic , number theory
Determine all pairs $(n,p)$ of nonnegative integers such that [list] [*] $p$ is a prime, [*] $n<2p$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

PEN A Problems, 53
Tags: divisibility theory
Suppose that $x, y,$ and $z$ are positive integers with $xy=z^2 +1$. Prove that there exist integers $a, b, c,$ and $d$ such that $x=a^2 +b^2$, $y=c^2 +d^2$, and $z=ac+bd$.

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, 26
Tags: induction , ratio , floor function , divisibility theory , inequalities
Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.

PEN A Problems, 63
Tags: pigeonhole principle , divisibility theory , induction
There is a large pile of cards. On each card one of the numbers $1$, $2$, $\cdots$, $n$ is written. It is known that the sum of all numbers of all the cards is equal to $k \cdot n!$ for some integer $k$. Prove that it is possible to arrange cards into $k$ stacks so that the sum of numbers written on the cards in each stack is equal to $n!$.

PEN A Problems, 3
Tags: function , polynomial , vieta , divisibility theory , algebra , induction
Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.

PEN A Problems, 12
Tags: divisibility theory
Let $k,m,$ and $n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_{s}=s(s+1).$ Prove that the product \[(c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k})\] is divisible by the product $c_{1}c_{2}\cdots c_{n}$.

PEN A Problems, 115
Tags: divisibility theory
Does there exist a $4$-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of $1992$?

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, 52
Tags: modular arithmetic , quadratic , 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, 99
Tags: divisibility theory
Let $n \ge 2$ be a positive integer, with divisors \[1=d_{1}< d_{2}< \cdots < d_{k}=n \;.\] Prove that \[d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}\] is always less than $n^{2}$, and determine when it divides $n^{2}$.

PEN A Problems, 79
Tags: divisibility theory
Determine all pairs of integers $(a, b)$ such that \[\frac{a^{2}}{2ab^{2}-b^{3}+1}\] is a positive integer.

PEN A Problems, 88
Tags: divisibility theory , modular arithmetic , induction , function
Find all positive integers $n$ such that $9^{n}-1$ is divisible by $7^n$.