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, 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, 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, 87
Tags: modular arithmetic , divisibility theory
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.

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, 80
Tags: inequalities , algebra , polynomial , divisibility theory
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

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, 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, 30
Tags: divisibility theory
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.

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, 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, 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, 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, 101
Tags: divisibility theory
Find all composite numbers $n$ having the property that each proper divisor $d$ of $n$ has $n-20 \le d \le n-12$.

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, 118
Tags: function , divisibility theory
Determine the highest power of $1980$ which divides \[\frac{(1980n)!}{(n!)^{1980}}.\]

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, 27
Tags: binomial coefficient , modular arithmetic , divisibility theory
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, 46
Tags: divisibility theory
Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.

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

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 $.

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, 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, 22
Tags: polynomial , binomial theorem , divisibility theory , algebra
Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

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