Found problems: 721
2009 Germany Team Selection Test, 2
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.$
2017 Dutch IMO TST, 2
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.
Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)
1994 IMO, 6
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
1992 Romania Team Selection Test, 6
Let $m,n$ be positive integers and $p$ be a prime number.
Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.
2025 Poland - First Round, 5
Positive integers $a, b, n$ are given. Assume that $a$ and $n$ are even, $b$ is odd and the number $ab(a+b)^{n-1}$ is divisible by $a^n+b^n$. Prove that there exist a prime number $p$, such that $p^{n+1}$ divides $a^n+b^n$.
1988 Brazil National Olympiad, 1
Find all primes which are sum of two primes and difference of two primes.
2008 Tuymaada Olympiad, 8
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23.
[i]Author: K. Kokhas[/i]
2022 Romania Team Selection Test, 3
Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers [i]suitable[/i] if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property.
[i]Călin Popescu and Marian Andronache[/i]
2009 Princeton University Math Competition, 5
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 $c$.
2003 Tournament Of Towns, 1
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
2015 Junior Balkan MO, 1
Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\]
Proposed by Moldova
2008 Tuymaada Olympiad, 2
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?
[i]Author: L. Emelyanov[/i]
[hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]
2023 VN Math Olympiad For High School Students, Problem 6
Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$
a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number.
b) $x^{2^n}+1,$ with $n$ is a positive integer.
2007 Balkan MO Shortlist, N5
Let $p \geq 5$ be a prime and let
\begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*}
where $q_i$ are primes. Prove,
\begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}
2020 IMC, 6
Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$
1999 Irish Math Olympiad, 2
A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies:
$ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime;
$ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$.
Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$
2019 IFYM, Sozopol, 1
Let $p_1, p_2, p_3$, and $p$ be prime numbers. Prove that there exist $x,y\in \mathbb{Z}$ such that
$y^2\equiv p_1 x^4-p_1 p_2^2 p_3^2\, (mod\, p)$.
2019 Peru EGMO TST, 4
Consider the numbers from $1$ to $32$. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the $32$ numbers as: $(1, 2), (3,4),(5, 6), (7, 8),..., (27, 28),(29, 30), (31,32)$, we get the following list of $16$ numbers: $3,7,11,5,...,11,59,7$. where there are repetitions. The game continues in a similar way until in the end only one number remains. Determine the highest possible value from the number that remains at the end.
2016 Iran MO (3rd Round), 1
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:
$$q |(x+1)^p-x^p$$
If and only if $$q \equiv 1 \pmod p$$
2001 Bundeswettbewerb Mathematik, 4
Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.
2024 Switzerland - Final Round, 1
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]
2008 Argentina Iberoamerican TST, 2
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.
2021 Pan-African, 3
Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold:
$\bullet ~~a_1\ge 2$.
$\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$
$\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$
Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$
2021 Romania National Olympiad, 3
Let $n\ge 2$ be a positive integer such that the set of $n$th roots of unity has less than $2^{\lfloor\sqrt n\rfloor}-1$ subsets with the sum $0$. Show that $n$ is a prime number.
[i]Cristi Săvescu[/i]
2018 Dutch BxMO TST, 3
Let $p$ be a prime number.
Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.