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.

AND:
OR:
NO:

Found problems: 121

PEN A Problems, 71
Tags: function , Divisibility Theory
Determine all integers $n > 1$ such that \[\frac{2^{n}+1}{n^{2}}\] is an integer.

PEN A Problems, 102
Tags: Divisibility Theory
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$

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, 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, 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, 13
Tags: floor function , logarithms , inequalities , Divisibility Theory
Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

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, 70
Tags: induction , binomial coefficients , 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, 24
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, 106
Tags: floor function , Divisibility Theory
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ 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, 104
Tags: Divisibility Theory
A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.

PEN A Problems, 114
Tags: number theory , greatest common divisor , modular arithmetic , Divisibility Theory
What is the greatest common divisor of the set of numbers \[\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?\]

PEN A Problems, 110
Tags: blogs , 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, 45
Tags: induction , Divisibility Theory
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, 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, 63
Tags: induction , pigeonhole principle , Divisibility Theory
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, 96
Tags: integration , modular arithmetic , number theory , relatively prime , Divisibility Theory , pen
Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

PEN A Problems, 39
Tags: Divisibility Theory , pen
Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]

PEN A Problems, 35
Tags: modular arithmetic , Divisibility Theory
Let $p \ge 5$ be a prime number. Prove that there exists an integer $a$ with $1 \le a \le p-2$ such that neither $a^{p-1} -1$ nor $(a+1)^{p-1} -1$ is divisible by $p^2$.

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, 112
Tags: algebra , Vieta , induction , number theory , relatively prime , Divisibility Theory , pen
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, 38
Tags: floor function , Divisibility Theory
Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.

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, 8
Tags: quadratics , search , modular arithmetic , algebra , Divisibility Theory , pen
The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.