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, 3
Tags: algebra , polynomial , vieta , function , induction , divisibility theory
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, 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, 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, 94
Tags: algorithm , divisibility theory
Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.

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, 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, 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, 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, 27
Tags: modular arithmetic , divisibility theory , binomial coefficient
Show that the coefficients of a binomial expansion $(a+b)^n$ where $n$ is a positive integer, are all odd, if and only if $n$ is of the form $2^{k}-1$ for some positive integer $k$.

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 $p$?

PEN A Problems, 82
Tags: quadratic , function , inequalities , vieta , divisibility theory , vieta jumping
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: number theory , greatest common divisor , modular arithmetic , relatively prime , divisibility 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, 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, 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, 9
Tags: number theory , relatively prime , divisibility theory
Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

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, 91
Tags: divisibility theory
Determine all pairs $(a, b)$ of positive integers such that $ab^2+b+7$ divides $a^2 b+a+b$.

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, 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, 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, 42
Tags: modular arithmetic , divisibility theory
Suppose that $2^n +1$ is an odd prime for some positive integer $n$. Show that $n$ must be a power of $2$.

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, 86
Tags: inequalities , induction , divisibility theory
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

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, 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 \[n={d_{1}}^{2}+{d_{2}}^{2}+{d_{3}}^{2}+{d_{4}}^{2}.\]