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

2022 Romania EGMO TST, P4
Tags: prime number , number theory
Let $p\geq 3$ be an odd positive integer. Show that $p$ is prime if and only if however we choose $(p+1)/2$ pairwise distinct positive integers, we can find two of them, $a$ and $b$, such that $(a+b)/\gcd(a,b)\geq p.$

2007 South East Mathematical Olympiad, 3
Tags: prime number , ratio , number theory
Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

2018 IFYM, Sozopol, 5
Tags: prime number , number theory
Find the solutions in prime numbers of the following equation $p^4 + q^4 + r^4 + 119 = s^2 .$

1993 Irish Math Olympiad, 2
Tags: number theory , prime number
A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.

2015 JBMO Shortlist, NT5
Tags: prime number , modular arithmetic , number theory
Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$

2020 Iran RMM TST, 1
Tags: prime number , number theory
For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd. [i]Proposed by Navid Safaie[/i]

2023 Stars of Mathematics, 1
Tags: prime number , perfect square , number theory
Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.

2020 Iran Team Selection Test, 6
Tags: number theory , prime number
$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

2024 CIIM, 3
Tags: number theory , euler totient function , prime number , summation
Given a positive integer \(n\), let \(\phi(n)\) denote the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). Find all possible positive integers \(k\) for which there exist positive integers \(1 \leq a_1 < a_2 < \dots < a_k\) such that: \[ \left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024 \]

2020 Macedonian NationŠ°l Olympiad, 1
Tags: number theory , fermat's little theorem , gcd , divisibility , prime number
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.

2012 Irish Math Olympiad, 4
Tags: number theory , prime number
Let $x$ > $1$ be an integer. Prove that $x^5$ + $x$ + $1$ is divisible by at least two distinct prime numbers.

2008 IMAR Test, 2
Tags: analytic geometry , number theory , prime number
A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point. [b]Dan Schwarz[/b]

2006 National Olympiad First Round, 2
Tags: number theory , prime number
If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2002 Manhattan Mathematical Olympiad, 1
Tags: relatively prime , prime number , number theory
Famous French mathematician Pierre Fermat believed that all numbers of the form $F_n = 2^{2^n} + 1$ are prime for all non-negative integers $n$. Indeed, one can check that $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ are all prime. a) Prove that $F_5$ is divisible by $641$. (Hence Fermat was wrong.) b) Prove that if $k \ne n$ then $F_k$ and $F_n$ are relatively prime (i.e. they do not have any common divisor except $1$) (Notice: using b) one can prove that there are infinitely many prime numbers)

2005 China Team Selection Test, 1
Tags: number theory , prime divisor , prime number , divisibility , power of 2
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2005 USAMO, 1
Tags: induction , relatively prime , number theory , divisor , arrangement , prime number , combinatorics
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

2017 China Team Selection Test, 6
Tags: polynomial , algebra , number theory , prime number
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

2012 Turkey Team Selection Test, 3
Tags: number theory , prime number , modular arithmetic
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.

1958 Poland - Second Round, 1
Tags: number theory , prime number , prime
Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.

2013 IFYM, Sozopol, 3
Tags: number theory , power of number , divisibility , prime number
Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.

2009 Purple Comet Problems, 13
Tags: number theory , prime number
How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?

2025 Poland - First Round, 3
Tags: number theory , prime number
Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that $$n+gcd(n, k)=k.$$

2011 China National Olympiad, 3
Tags: number theory , algorithm , prime number , divisibility , modular arithmetic
Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

2019 Korea Junior Math Olympiad., 3
Tags: prime number , number theory
Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition: There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.

2016 Croatia Team Selection Test, Problem 4
Tags: decimal representation , prime , prime number , primitive root , number theory
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.