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: 364

2022 239 Open Mathematical Olympiad, 5
Tags: number theory , prime
Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$

1992 Romania Team Selection Test, 6
Tags: number theory , prime , integer polynomial , prime number
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.

2016 Thailand Mathematical Olympiad, 6
Tags: prime , power of 2 , number theory
Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$.

2021 Durer Math Competition Finals, 1
Tags: number theory , prime
Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.

2010 Thailand Mathematical Olympiad, 6
Tags: number theory , prime , divide
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

2025 Israel TST, P2
Tags: number theory , prime
Prove that for all primes \( p \) such that \( p \equiv 3 \pmod{4} \) or \( p \equiv 5 \pmod{8} \), there exist integers \[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \] such that \[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]

1997 Israel National Olympiad, 5
Tags: coprime , prime , number theory
The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

2011 Brazil Team Selection Test, 2
Tags: number theory , prime
Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.

2019 Indonesia MO, 1
Tags: number theory , prime
Given that $n$ and $r$ are positive integers. Suppose that \[ 1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r) \] Prove that $n$ is a composite number.

2023 Brazil Team Selection Test, 3
Tags: number theory , prime factorization , function , prime number , prime
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2015 May Olympiad, 4
Tags: combinatorics , prime , number theory
The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.

1997 Israel Grosman Mathematical Olympiad, 1
Tags: number theory , digit , prime
Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

2013 VJIMC, Problem 1
Tags: number theory , prime
Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.

2017 Iran MO (3rd round), 1
Tags: number theory , divisibility , prime number , prime
Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

2022 Indonesia TST, N
Tags: infinity , polynomial , number theory , prime , product
Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

2014 Estonia Team Selection Test, 1
Tags: combinatorics , prime
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

2007 Estonia Team Selection Test, 3
Tags: number theory , power of prime , prime
Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

2000 Singapore Team Selection Test, 2
Tags: number theory , prime , perfect square
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square

2021 Romanian Master of Mathematics Shortlist, N2
Tags: number theory , combinatorial number theory , prime
We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.

2022 IMO Shortlist, N2
Tags: number theory , prime , divisibility
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2019 India PRMO, 21
Tags: number theory , prime number , set , prime
Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

2025 International Zhautykov Olympiad, 3
Tags: number theory , construction , prime
A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

2019 Bundeswettbewerb Mathematik, 4
Tags: number theory , prime
Prove that for no integer $k \ge 2$, between $10k$ and $10k + 100$ there are more than $23$ prime numbers.

2016 Costa Rica - Final Round, N2
Tags: prime , number theory
Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.

2019 Saudi Arabia BMO TST, 1
Tags: number theory , divide , divisible , prime
Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$