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: 715

2023 Germany Team Selection Test, 3
Tags: algebra , polynomial , number theory , prime numbers , algebra proposed
Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\] is at most $\binom{2023}{2}$ times divisible by $p$. [i]Proposed by Ashwin Sah[/i]

2005 Slovenia National Olympiad, Problem 2
Tags: number theory , prime numbers
For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?

2016 Iran Team Selection Test, 3
Tags: number theory , prime numbers , modular arithmetic , Quadratic Residues
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

1987 AMC 8, 9
Tags: number theory , least common multiple , relatively prime , prime numbers
When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is $\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$

1977 IMO Shortlist, 10
Tags: number theory , prime numbers , prime factorization , Dirichlet s Theorem , IMO , IMO 1977
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 2
Tags: number theory , prime numbers
Let $p$ be a prime number and $F=\left \{0,1,2,...,p-1 \right \}$. Let $A$ be a proper subset of $F$ that satisfies the following property: if $a,b \in A$, then $ab+1$ (mod $p$) $ \in A$. How many elements can $A$ have? (Justify your answer.)

2014 Saudi Arabia BMO TST, 1
Tags: function , number theory , prime numbers , number theory unsolved
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

2016 IFYM, Sozopol, 3
Tags: number theory , prime numbers
Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.

2018 Latvia Baltic Way TST, P13
Tags: number theory , number theory unsolved , prime numbers
Determine whether there exists a prime $q$ so that for any prime $p$ the number $$\sqrt[3]{p^2+q}$$ is never an integer.

Russian TST 2014, P1
Tags: number theory , Divisibility , prime numbers
Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$

2018 Hanoi Open Mathematics Competitions, 3
Tags: number theory , prime numbers
Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $4x^2 + 8y^2 + (2x-3y)p-12xy = 0$. Which below number is a perfect square number for every such triple $(x,y, p)$? A. $4y + 1$ B. $2y + 1$ C. $8y + 1$ D. $5y - 3$ E. $8y - 1$

2019 Hong Kong TST, 1
Tags: number theory , prime numbers
Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation \[ p_{n+2} = p_{n+1} + p_n + k \] holds for all positive integers $n$.

2020 June Advanced Contest, 2
Tags: graph theory , complete subgraph , combinatorics , number theory , prime numbers
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?

2018 VJIMC, 2
Tags: number theory , prime numbers
Find all prime numbers $p$ such that $p^3$ divides the determinant \[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]

2017 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: number theory , prime numbers
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares

2013 Puerto Rico Team Selection Test, 3
Tags: number theory , prime numbers
Find all pairs of natural numbers n and prime numbers p such that $\sqrt{n+\frac{p}{n}}$ is a natural number.

1992 Romania Team Selection Test, 6
Tags: number theory , prime , Integer Polynomial , prime numbers
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 IberoAmerican, 1
Tags: number theory , prime numbers , Iberoamerican , Iberoamerican 2016
Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$

2006 Germany Team Selection Test, 1
Tags: number theory , prime numbers , number theory proposed
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?

1982 Bulgaria National Olympiad, Problem 1
Tags: inequalities , number theory , prime numbers
Find all pairs of natural numbers $(n,k)$ for which $(n+1)^{k}-1 = n!$.

2009 Bosnia and Herzegovina Junior BMO TST, 3
Tags: number theory , prime , Division , prime numbers
Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

2022 IMC, 6
Tags: number theory , prime numbers , modular arithmetic , IMC 2022
Let $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$.

2011 Danube Mathematical Competition, 3
Tags: number theory , prime numbers , divisible
Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.

2006 Turkey Team Selection Test, 1
Tags: induction , quadratics , Gauss , modular arithmetic , number theory , prime numbers , number theory proposed
For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

KoMaL A Problems 2017/2018, A. 717
Tags: number theory , prime numbers , Perfect Squares
Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$. Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.