Found problems: 387
A set of $2003$ positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.
Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.
Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?
Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.
Prove the theorem:
There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.
$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.
Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.
Find all positive integers $n$ for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.
Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?