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

2021 Argentina National Olympiad, 5
Tags: number theory , prime number
The sequence $a_n (n\geq 1)$ of natural numbers is defined as $a_{n+1}=a_n+b_n,$ where $b_n$ is the number that has the same digits as $a_n$ but in the opposite order ($b_n$ can start with $0$). For example, if $a_1=180,$ then $a_2=261, a_3=423.$ a) Decide if $a_1$ can be chosen so that $a_7$ is prime. b) Decide if $a_1$ can be chosen so that $a_5$ is prime.

2003 China National Olympiad, 2
Tags: combinatorics , number theory , prime number
Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$; ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$; iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$. [i]Yao Jiangang[/i]

2004 France Team Selection Test, 3
Tags: number theory , prime number , induction
Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

2025 Bangladesh Mathematical Olympiad, P4
Tags: number theory , prime number , diophantine equation , bounding , modular arithmetic
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

2023 India Regional Mathematical Olympiad, 2
Tags: number theory , prime number
Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

2011 Dutch IMO TST, 4
Tags: number theory , sequence , prime number , recurrence relation
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

1987 AMC 8, 9
Tags: number theory , least common multiple , relatively prime , prime number
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 $\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$

2009 Iran Team Selection Test, 2
Tags: number theory , inequalities , pigeonhole principle , prime number , diophantine equation , modular arithmetic
Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

1992 Spain Mathematical Olympiad, 4
Tags: arithmetic sequence , prime number , number theory
Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.

2017 CentroAmerican, 3
Tags: prime number , number theory
Tita the Frog sits on the number line. She is initially on the integer number $k>1$. If she is sitting on the number $n$, she hops to the number $f(n)+g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distinct integers.

2013 All-Russian Olympiad, 3
Tags: number theory , prime number
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).

2024 Francophone Mathematical Olympiad, 4
Tags: number theory , prime number , divisor
Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

2013 JBMO TST - Turkey, 2
Tags: number theory , modular arithmetic , prime number
[b]a)[/b] Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares. [b]b)[/b] Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?

1999 China Team Selection Test, 2
Tags: prime number , number theory
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2007 All-Russian Olympiad Regional Round, 8.3
Tags: number theory , prime number
Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}\minus{}1,p_{3}|p_{2}^{2}\minus{}1,...,p_{1}|p_{2007}^{2}\minus{}1$.

PEN E Problems, 24
Tags: limit , inequalities , logarithm , prime number , number theory , floor function
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

2007 Tuymaada Olympiad, 4
Tags: logarithm , number theory , limit , greatest common divisor , prime number , probability , ratio
Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

2014 AMC 8, 23
Tags: prime number , number theory
Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of you two uniform numbers is today's date. What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$

2020 MMATHS, I3
Tags: prime number , number theory
Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$. Find $p + q + r$. [i]Proposed by Andrew Wu[/i]

2018 Malaysia National Olympiad, A2
Tags: number theory , prime number
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 $a$.

2007 Silk Road, 1
Tags: number theory , prime number
On the board are written $2 , 3 , 5 ,... , 2003$ , that is, all the prime numbers of the interval $[2,2007]$ . The operation of [i]simplification [/i] is the replacement of two numbers $a , b$ by a maximal prime number not exceeding $\sqrt{a^2-a b+b^2}$ . First, the student erases the number $q, 2<q<2003$, then applies the [i]simplification [/i] operation to the remaining numbers until one number remains. Find the maximum possible and minimum possible values of the number obtained in the end. How do these values depend on the number $q$?

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

1955 Miklós Schweitzer, 4
Tags: number theory , prime number
[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

2016 Junior Regional Olympiad - FBH, 3
Tags: number theory , prime number
Prove that when dividing a prime number with $30$, remainder is always not a composite number

2011 All-Russian Olympiad Regional Round, 9.7
Tags: number theory , prime number
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)