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

2013 All-Russian Olympiad, 3
Tags: quadratic , modular arithmetic , prime number , number theory
Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. [i]V. Senderov[/i]

1974 Miklós Schweitzer, 4
Tags: superior algebra , ring theory , function , induction , number theory , prime number
Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

2024 Romanian Master of Mathematics, 2
Tags: algorithm , number theory , prime number , partial sums , permutation
Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$. [i]Will Steinberg, United Kingdom[/i]

2011 Greece Team Selection Test, 1
Tags: prime number , number theory
Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

2014 Iran Team Selection Test, 2
Tags: function , prime number , number theory
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $i) \exists n\in \mathbb{N}:f(n)\neq n$ $ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$

2007 Thailand Mathematical Olympiad, 18
Tags: prime , remainder , number theory , prime number
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

2008 Tuymaada Olympiad, 2
Tags: prime number , number theory
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2010 India IMO Training Camp, 8
Tags: prime number , number theory
Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

2023 Brazil EGMO Team Selection Test, 2
Tags: number theory , prime number , set
Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions: $(i)$ the prime factors of any element of $B$ are in $A$; $(ii)$ no term of $B$ divides another element of this set.

2011 Danube Mathematical Competition, 3
Tags: number theory , prime number , 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 $n$.

2023 Singapore Junior Math Olympiad, 4
Tags: number theory , prime number
Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.

2014 Contests, 1
Tags: prime number , number theory
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

2006 MOP Homework, 4
Tags: number theory , prime number
Given a prime number $p > 2$. Find the least $n\in Z_+$, for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\ (\text{mod}\ p)$

the 11th XMO, 3
Tags: prime number , number theory
Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

2013 Peru IMO TST, 5
Tags: binomial coefficient , number theory , prime number
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2014 Junior Regional Olympiad - FBH, 4
Tags: number theory , prime number
Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

2022 Kosovo National Mathematical Olympiad, 3
Tags: number theory , prime number
Let $a,b$ and $c$ be positive integers such that $a!+b+c,b!+c+a$ and $c!+a+b$ are prime numbers. Show that $\frac{a+b+c+1}{2}$ is also a prime number.

1991 IMTS, 1
Tags: number theory , prime number
Use each of the digits 1,2,3,4,5,6,7,8,9 exactly twice to form distinct prime numbers whose sum is as small as possible. What must this minimal sum be? (Note: The five smallest primes are 2,3,5,7, and 11)

2008 Romania Team Selection Test, 2
Tags: prime number , number theory
Are there any sequences of positive integers $ 1 \leq a_{1} < a_{2} < a_{3} < \ldots$ such that for each integer $ n$, the set $ \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\}$ contains finitely many prime numbers?

1997 Korea National Olympiad, 4
Tags: number theory , prime number
For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

2018 AMC 10, 11
Tags: prime number
Which of the following expressions is never a prime number when $p$ is a prime number? $\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$

2017 Bundeswettbewerb Mathematik, 4
Tags: sequence , recursion , prime number , integer , number theory
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$: (a) The number $a_n$ is a positive integer. (b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$. (c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.

2018 Malaysia National Olympiad, A2
Tags: prime number , number theory
Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

1981 Putnam, B3
Tags: prime number , number theory
Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$

1994 China Team Selection Test, 2
Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.