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

1995 IMO Shortlist, 8
Tags: number theory , prime number , inequality , minimization
Let $ p$ be an odd prime. Determine positive integers $ x$ and $ y$ for which $ x \leq y$ and $ \sqrt{2p} \minus{} \sqrt{x} \minus{} \sqrt{y}$ is non-negative and as small as possible.

1989 IMO Longlists, 93
Tags: modular arithmetic , number theory , prime number , sequence , power of number
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2025 Polish MO Finals, 2
Tags: number theory , prime number
Positive integers $k, m, n ,p $ integers are such that $p=2^{2^n}+1$ is prime and $p\mid 2^k-m$. Prove that there exists a positive integer $l$ such that $p^2\mid 2^l-m$.

2021/2022 Tournament of Towns, P1
Tags: number theory , prime number
Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

1993 Irish Math Olympiad, 2
Tags: prime number , number theory
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.

1978 IMO Longlists, 10
Tags: prime number , arithmetic sequence , number theory
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.

1998 India Regional Mathematical Olympiad, 2
Tags: modular arithmetic , number theory , prime number
Let $n$ be a positive integer and $p_1, p_2, p_3, \ldots p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2$. prove that $6$ divides $n$.

2024 Turkey EGMO TST, 5
Tags: grid , combinatorics , prime number
Let $p$ be a given prime number. For positive integers $n,k\geq2$ let $S_1, S_2,\dots, S_n$ be unit square sets constructed by choosing exactly one unit square from each of the columns from $p\times k$ chess board. If $|S_i \cap S_j|=1$ for all $1\leq i < j \leq n$ and for any duo of unit squares which are located at different columns there exists $S_i$ such that both of these unit squares are in $S_i$ find all duos of $(n,k)$ in terms of $p$. Note: Here we denote the number of rows by $p$ and the number of columns by $k$.

2018 Korea Winter Program Practice Test, 4
Tags: number theory , prime number
Let $p=4k+1$ be a prime. $S$ is a set of all possible residues equal or smaller then $2k$ when $\frac{1}{2} \binom{2k}{k} n^k$ is divided by $p$. Show that \[ \sum_{x \in S} x^2 =p \]

2021 Kosovo National Mathematical Olympiad, 4
Tags: algebra , polynomial , number theory , prime number
Let $P(x)$ be a polynomial with integer coefficients. We will denote the set of all prime numbers by $\mathbb P$. Show that the set $\mathbb S := \{p\in\mathbb P : \exists\text{ }n \text{ s.t. }p\mid P(n)\}$ is finite if and only if $P(x)$ is a non-zero constant polynomial.

2006 Estonia Team Selection Test, 6
Tags: number theory , prime number , prime factorization
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.

2005 China Team Selection Test, 1
Tags: prime divisor , number theory , 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.

2020 June Advanced Contest, 2
Tags: graph theory , complete subgraph , combinatorics , number theory , prime number
Let $p$ be a prime number. At a school of $p^{2020}$ students it is required that each club consist of exactly $p$ students. Is it possible for each pair of students to have exactly one club in common?

2009 Jozsef Wildt International Math Competition, W. 5
Tags: number theory , prime number
Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.

2011 All-Russian Olympiad Regional Round, 9.7
Tags: prime number , number theory
Find all prime numbers $p$, $q$ and $r$ such that the fourth power of any of them minus one is divisible by the product of the other two. (Author: V. Senderov)

2024 Macedonian Balkan MO TST, Problem 3
Tags: number theory , prime number
Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.

2017 Romania National Olympiad, 4
Tags: number theory , prime , digit , prime number
Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

2018 Turkey Team Selection Test, 1
Tags: quadratic , polynomial , number theory , prime number , c.r.t
Prove that, for all integers $a, b$, there exists a positive integer $n$, such that the number $n^2+an+b$ has at least $2018$ different prime divisors.

2007 Macedonia National Olympiad, 3
Tags: prime number , number theory
Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy \[a | b+c+bc, b | c+a+ca, c | a+b+ab.\] Prove that at least one of the numbers $a, b, c$ is not prime.

Kvant 2020, M2597
Tags: number theory , prime number
Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$. [i]Proposed by M. Antipov[/i]

2023 Kazakhstan National Olympiad, 5
Tags: number theory , prime number
Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$

2021 Azerbaijan EGMO TST, 1
Tags: prime number , prime , number theory
p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

2016 Hong Kong TST, 4
Tags: number theory , prime number
Find all triples $(m,p,q)$ such that \begin{align*} 2^mp^2 +1=q^7, \end{align*} where $p$ and $q$ are ptimes and $m$ is a positive integer.

2024 Kazakhstan National Olympiad, 2
Tags: number theory , prime number , function
Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]

2003 Austrian-Polish Competition, 9
Tags: pigeonhole principle , number theory , prime number , combinatorics
Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square. [hide] I think that the upper limit for such subset is 37.[/hide]