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 IMC, 4
Tags: algebra , polynomial , number theory , modular arithmetic , prime numbers , IMC , IMC 2023
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

2023 Romanian Master of Mathematics, 1
Tags: number theory , prime numbers , RMM 2023 , Diophantine equation
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

2009 Germany Team Selection Test, 1
Tags: number theory , prime numbers , number theory unsolved
For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2019 Brazil Team Selection Test, 4
Tags: number theory , prime numbers , Sets , Sum of Squares
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.

2013 All-Russian Olympiad, 3
Tags: number theory , prime numbers , number theory proposed
Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one).

2021 Regional Olympiad of Mexico Southeast, 2
Tags: number theory , prime numbers , arithmetic sequence
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.

2009 Turkey Team Selection Test, 1
Tags: algebra , polynomial , abstract algebra , number theory , prime numbers , algebra unsolved
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

2000 AMC 12/AHSME, 6
Tags: number theory , prime numbers
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

2004 Irish Math Olympiad, 1
Tags: number theory , prime numbers
Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.

2014 JBMO Shortlist, 4
Tags: number theory , prime numbers , AZE JBMO TST
Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2021 South East Mathematical Olympiad, 2
Tags: number theory , prime numbers
Let $p\geq 5$ be a prime number, and set $M=\{1,2,\cdots,p-1\}.$ Define $$T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}.$$ If $\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p},$ with $0\leq k\leq p-1,$ where $\left[\alpha\right]$ denotes the largest integer that does not exceed $\alpha,$ determine the value of $k.$

2010 Singapore MO Open, 3
Tags: ratio , number theory , prime numbers , arithmetic sequence , number theory proposed
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

2022 Poland - Second Round, 3
Tags: number theory , prime numbers , modular congruences , Divisibility
Positive integers $a,b,c$ satisfying the equation $$a^3+4b+c = abc,$$ where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.

2017 Iran Team Selection Test, 4
Tags: number theory , Iran , Iranian TST , prime numbers , modular arithmetic
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

Russian TST 2015, P2
Tags: number theory , prime numbers
Let $p\geqslant 5$ be a prime number. Prove that the set $\{1,2,\ldots,p - 1\}$ can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo $p{}$.

1989 IMO Longlists, 7
Tags: floor function , number theory , prime numbers , algebra unsolved , algebra
For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.

2003 Poland - Second Round, 4
Tags: number theory , combinatorics , Combinatorial Number Theory , Poland , pigeonhole principle , prime numbers
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.

2011 All-Russian Olympiad, 3
Tags: modular arithmetic , number theory , relatively prime , prime numbers , number theory proposed
For positive integers $a>b>1$, define \[x_n = \frac {a^n-1}{b^n-1}\] Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers. [i]V. Senderov[/i]

2021 ITAMO, 6
Tags: number theory , prime numbers
A sequence $x_1, x_2, ..., x_n, ...$ consists of an initial block of $p$ positive distinct integers that then repeat periodically. This means that $\{x_1, x_2, \dots, x_p\}$ are $p$ distinct positive integers and $x_{n+p}=x_n$ for every positive integer $n$. The terms of the sequence are not known and the goal is to find the period $p$. To do this, at each move it possible to reveal the value of a term of the sequence at your choice. (a) Knowing that $1 \le p \le 10$, find the least $n$ such that there is a strategy which allows to find $p$ revealing at most $n$ terms of the sequence. (b) Knowing that $p$ is one of the first $k$ prime numbers, find for which values of $k$ there exist a strategy that allows to find $p$ revealing at most $5$ terms of the sequence.

2021 Kyiv City MO Round 1, 11.5
Tags: number theory , prime numbers
For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

2012-2013 SDML (High School), 5
Tags: number theory , prime numbers , SDML High School , 2012-13 SDML Round 2
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

1995 IMO Shortlist, 8
Tags: number theory , prime numbers , Inequality , minimization , IMO Shortlist
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.

2019 Centers of Excellency of Suceava, 1
Tags: number theory , prime numbers
Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two. [i]Gherghe Stoica[/i]

2018 Dutch BxMO TST, 3
Tags: number theory , prime numbers , Divisibility
Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

2014 Ukraine Team Selection Test, 9
Tags: number theory , prime numbers , parameterization , Diophantine Equations , diophantine , system of equations
Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.