Found problems: 7
Using only the ruler, divide the side of a square table into $n$ equal parts.
All lines drawn must lie on the surface of the table.
Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$
For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$
Set $S=\{1,2,...,n\}$ is firstly divided on $m$ disjoint nonempty subsets, and then on $m^2$ disjoint nonempty subsets. Prove that some $m$ elements of set $S$ were after first division in same set, and after the second division were in $m$ different sets
Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$?
By dividing the integer $m$ by the integer $n, 22$ is the quotient and $5$ the remainder.
As the division of the remainder with $n$ continues, the new quotient is $0.4$ and the new remainder is $0.2$.
Find $m$ and $n$.
Alex writes the sixteen digits $2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9$ side by side in any order and then places a colon somewhere between two digits, so that a division task arises. Can the result of this calculation be $2$?