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, 70
Tags: induction , binomial coefficient , 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, 88
Tags: induction , function , modular arithmetic , divisibility theory
Find all positive integers $n$ such that $9^{n}-1$ is divisible by $7^n$.

2017 Pan-African Shortlist, N1
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, 55
Tags: divisibility theory
Show that for every natural number $n$ the product \[\left( 4-\frac{2}{1}\right) \left( 4-\frac{2}{2}\right) \left( 4-\frac{2}{3}\right) \cdots \left( 4-\frac{2}{n}\right)\] is an integer.

PEN A Problems, 13
Tags: floor function , inequalities , divisibility theory , logarithm
Show that for all prime numbers $p$, \[Q(p)=\prod^{p-1}_{k=1}k^{2k-p-1}\] is an integer.

PEN A Problems, 14
Tags: modular arithmetic , number theory , divisibility theory
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.

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, 6
Tags: algebra , polynomial , vieta , divisibility theory
[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

PEN A Problems, 17
Tags: modular arithmetic , quadratic , number theory , relatively prime , divisibility theory
Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.

PEN A Problems, 111
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, 41
Tags: modular arithmetic , divisibility theory
Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.

PEN A Problems, 112
Tags: algebra , vieta , induction , number theory , relatively prime , divisibility 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, 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, 116
Tags: divisibility theory
What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits $2$ through $9$?

PEN A Problems, 72
Tags: modular arithmetic , number theory , divisibility 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, 36
Tags: floor function , divisibility theory
Let $n$ and $q$ be integers with $n \ge 5$, $2 \le q \le n$. Prove that $q-1$ divides $\left\lfloor \frac{(n-1)!}{q}\right\rfloor $.

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, 57
Tags: divisibility theory
Prove that for every $n \in \mathbb{N}$ the following proposition holds: $7|3^n +n^3$ if and only if $7|3^{n} n^3 +1$.

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, 51
Tags: divisibility theory
Let $a,b,c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^{k}$ and $b+c=2^{m}$ for some integers $k$ and $m$, then $a=1$.

PEN A Problems, 34
Tags: divisibility theory
Let $p_{1}, p_{2}, \cdots, p_{n}$ be distinct primes greater than $3$. Show that \[2^{p_{1}p_{2}\cdots p_{n}}+1\] has at least $4^{n}$ divisors.

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, 32
Tags: floor function , divisibility theory
Let $ a$ and $ b$ be natural numbers such that \[ \frac{a}{b}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}. \] Prove that $ a$ is divisible by $ 1979$.

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, 8
Tags: quadratic , search , modular arithmetic , divisibility theory , algebra
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 $a=0$.