This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 721

2021 Saint Petersburg Mathematical Olympiad, 1
Tags: number theory , prime number
Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum. [i]A. Khrabov[/i]

PEN A Problems, 47
Tags: floor function , number theory , factorial , prime number , divisibility theory
Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.

2005 Iran MO (3rd Round), 5
Tags: induction , prime number , number theory
Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.

2011 Math Prize for Girls Olympiad, 3
Tags: modular arithmetic , prime number , number theory
Let $n$ be a positive integer such that $n + 1$ is divisible by 24. Prove that the sum of all the positive divisors of $n$ is divisible by 24.

2022 Austrian MO National Competition, 4
Tags: prime number , diophantine equation , diophantine , number theory
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

LMT Speed Rounds, 2016.6
Tags: number theory , prime number
A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers. [i]Proposed by Evan Fang

2019 IMEO, 4
Tags: algebra , polynomial , prime number
Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well. [i]Proposed by Valentio Iverson (Indonesia)[/i]

2013 Singapore Junior Math Olympiad, 3
Tags: prime number , number theory
Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.

2018 Brazil Team Selection Test, 2
Tags: relatively prime , prime number , number theory
Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$. [i](R. Salimov)[/i]

2012 Belarus Team Selection Test, 1
Tags: algorithm , number theory , prime number , divisibility , number of divisors
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

2011 Putnam, B2
Tags: algebra , polynomial , number theory , prime number , rational root theorem
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$

2011 China Western Mathematical Olympiad, 1
Tags: modular arithmetic , prime number , number theory
Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?

2006 Estonia Team Selection Test, 6
Tags: prime number , prime factorization , number theory
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

1959 Miklós Schweitzer, 1
Tags: number theory , real analysis , prime number
[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ [b](N.17)[/b]

2017 IFYM, Sozopol, 1
Tags: algebra , number theory , prime number
Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

2012 BMT Spring, 7
Tags: prime number , number theory
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $, $ a $, $ b $, $ c $, $ d \neq 3 $ , and $ d \le 50 $. Find the maximum value of the difference between two of these prime numbers.

1998 IMO, 6
Tags: number theory , functional equation , prime number , algebra
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

2016 Mathematical Talent Reward Programme, MCQ: P 8
Tags: prime number , number theory
Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is [list=1] [*] 283 [*] 307 [*] 593 [*] 691 [/list]

2022 AIME Problems, 5
Tags: number theory , prime number
Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.

2009 Princeton University Math Competition, 3
Tags: number theory , prime number
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 $c$.

2020 Peru IMO TST, 1
Tags: number theory , perfect power , prime number
Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2025 6th Memorial "Aleksandar Blazhevski-Cane", P1
Tags: number theory , prime number , diophantine equation
Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

2019 Macedonia Junior BMO TST, 1
Tags: number theory , prime number
Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number.

2012 Morocco TST, 1
Tags: number theory , prime number
Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

2018 AMC 10, 5
Tags: number theory , prime number
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$