Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$?

Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).

Lets say that a positive integer is $good$ if its equal to the the subtraction of two positive integer cubes. For example: $7$ is a $good$ prime because $2^3-1^3=7$. Determine how much the last digit of a $good$ prime may be worth. Give all the possibilities.

Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.

Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?

Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.

Let $p$ be an odd prime number. Ana and Ben are playing a game with alternate moves as follows: in each move, the player which has the turn choose a number, which was not choosen before by any of the player, from the set $\{1,2,...,2p-3,2p-2\}$. This process continues until no number is left. After the end of the process, each player create the number by taking the product of the choosen numbers and then add 1. We say a player wins if the number that did create is divisible by $p$, while the number that did create the opponent it is not divisible by $p$, otherwise we say the game end in a draw. Ana start first move. Does it exist a strategy for any of the player to win the game? [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

The sequence $ < x_n >$ is defined through: $ x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1$ for $ n > 0$ Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.

Is the following set of prime numbers $p$ finite or infinite, where each $p$ [b]doesn't[/b] divide the numbers that can be expressed as $n^{2016}+2016^{2016}$ for $n\in \mathbb{N}$, if: a) $p=4k+3$; b) $p=4k+1$?

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

Let $A$ be a finite subset of prime numbers and $a> 1$ be a positive integer. Show that the number of positive integers $m$ for which all prime divisors of $a^m-1$ are in $A$ is finite.

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.

For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

Alice is once again very bored in class. On a whim, she chooses three primes $p$, $q$, $r$ independently and uniformly at random from the set of primes at most 30. She then calculates the roots of $px^2+qx+r$. What is the probability that at least one of her roots is an integer?

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $

Find all pairs $(n, p)$ that satisfy the following condition, where $n$ is a positive integer and $p$ is a prime number. [b]Condition)[/b] $2n-1$ is a divisor of $p-1$ and $p$ is a divisor of $4n^2+7$.

Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.

Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ($p_1=2$, $p_2=3$, $p_3=3$ etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$. Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$. [hide=example]For $k=2$ $p_1=2$, $p_2=3$, $N=6$. So in set $\{ 1,2,3,4,5,6 \}$ we can find $3$ number satisfying thesis: $2$, $3$ and $4$. ($1$ and $5$ are not divisible by $2$ or $3$, and $6$ is divisible by both of them so by even number of primes )[/hide]

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$