Found problems: 1048
Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.
Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.
If $a$ and $b$ are positive integers, we say that $a$ [i]almost divides[/i] $b$ if $a$ divides at least one of $b - 1$ and $b + 1$. We call a positive integer $n$ [i]almost prime[/i] if the following holds: for any positive integers $a, b$ such that $n$ almost divides $ab$, we have that $n$ almost divides at least one of $a$ and $b$. Determine all almost prime numbers.
[hide = original link][url]https://mathematical.olympiad.ch/fileadmin/user_upload/Archiv/Intranet/Olympiads/Mathematics/deploy/exams/2024/FinalRound/Exam/englishFinalRound2024.pdf[/url]!![/hide]
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.
For a prime number $p > 3$, define the following irreducible fraction:
$$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$
Prove that $m$ is divisible by $p$.
[i]Proposed by Oleksii Masalitin[/i]
For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$
$\textbf{(D) }n-1\qquad
\textbf{(E) }n$
All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$
Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.
Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$
Find all prime numbers $p,q$ such that:
$$p^4+p^3+p^2+p=q^2+q$$
On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.
Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation
\[ p_{n+2} = p_{n+1} + p_n + k \]
holds for all positive integers $n$.
Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .
Let $\mathbb{Z^+}$ and $\mathbb{P}$ denote the set of positive integers and the set of prime numbers, respectively. A set $A$ is called $S-\text{proper}$ where $A, S \subset \mathbb{Z^+}$ if there exists a positive integer $N$ such that for all $a \in A$ and for all $0 \leq b <a$ there exist $s_1, s_2, \ldots, s_n \in S$ satisfying $ b \equiv s_1+s_2+\cdots+s_n \pmod a$ and $1 \leq n \leq N.$
Find a subset $S$ of $\mathbb{Z^+}$ for which $\mathbb{P}$ is $S-\text{proper}$ but $\mathbb{Z^+}$ is not.
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.
Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that
[list=1]
[*] $A\cap B=\{1\};$
[*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$;
[*] each prime number is a divisor of some number in $A$ and also some number in $B$;
[*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$.
[*] Each set has infinitely many composite numbers.
[/list]
Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent:
(A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$
(B) The number $n$ has at least two different prime divisors.