Found problems: 121
Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.
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 $p^{2}$.
Determine all pairs $(x, y)$ of positive integers with $y \vert x^2 +1$ and $x^2 \vert y^3 +1$.
Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.
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.
Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.
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]
Find all positive integers $a$ and $b$ such that \[\frac{a^{2}+b}{b^{2}-a}\text{ and }\frac{b^{2}+a}{a^{2}-b}\] are both integers.
Determine all triples of positive integers $(a, m, n)$ such that $a^m +1$ divides $(a+1)^n$.
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 \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\]
Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.
Determine all integers $n > 1$ such that \[\frac{2^{n}+1}{n^{2}}\] is an integer.
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.
Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.
Find all composite numbers $n$ having the property that each proper divisor $d$ of $n$ has $n-20 \le d \le n-12$.
Find all positive integers, representable uniquely as \[\frac{x^{2}+y}{xy+1},\] where $x$ and $y$ are positive integers.
Determine all ordered pairs $(m, n)$ of positive integers such that \[\frac{n^{3}+1}{mn-1}\] is an integer.
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.
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 $9$?
Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.
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 $p^2$.
Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]
For which positive integers $k$, is it true that there are infinitely many pairs of positive integers $(m, n)$ such that \[\frac{(m+n-k)!}{m! \; n!}\] is an integer?