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, 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, 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.

1998 Romania Team Selection Test, 3
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, 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]

2018 Moldova Team Selection Test, 12
Tags: floor function , modular arithmetic , Divisibility Theory
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

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, 47
Tags: factorial , floor function , number theory , prime numbers , 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, 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, 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, 107
Tags: logarithms , inequalities , rearrangement inequality , Divisibility Theory
Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.

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 \[p(n)<p(n+1)<p(n+2).\]

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, 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, 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, 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, 4
Tags: search , algebra , polynomial , Vieta , symmetry , quadratics , Divisibility Theory
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.

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, 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, 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, 3
Tags: algebra , polynomial , Vieta , function , induction , Divisibility Theory , pen
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, 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, 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, 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, 62
Tags: Divisibility Theory
Let $p(n)$ be the greatest odd divisor of $n$. Prove that \[\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.\]

PEN A Problems, 85
Tags: Divisibility Theory
Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.