Found problems: 5
Given is a grid with $2$ rows and $120$ columns, such that each cell has a number from the set $1, 2, ..., 120$. It is known that in each column, the upper number in it is smaller than the lower number, and in each row, the numbers are in non-strict increasing order from left to right. Prove that the number of these tables is multiple of $239$.
In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom.
Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$.
([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape)
[i]Proposed by Morteza Saghafian[/i]
In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom.
Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$.
([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape)
[i]Proposed by Morteza Saghafian[/i]
Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set
\[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\]
are divisible by $p$.
[i]Tristan Shin[/i]
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.