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

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 $n$

2018 Mediterranean Mathematics OIympiad, 3
Tags: prime number , number theory
An integer $a\ge1$ is called [i]Aegean[/i], if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$. (Proposed by Gerhard Woeginger, Austria)

2023 Turkey EGMO TST, 2
Tags: prime number , number theory
Find all pairs of $p,q$ prime numbers that satisfy the equation $$p(p^4+p^2+10q)=q(q^2+3)$$

2005 IMO Shortlist, 5
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.

2014 AMC 8, 4
Tags: modular arithmetic , number theory , prime number
The sum of two prime numbers is $85$. What is the product of these two prime numbers? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

2016 Hong Kong TST, 4
Tags: prime number , number theory
Mable and Nora play a game according to the following steps in order: 1. Mable writes down any 2015 distinct prime numbers in ascending order in a row. The product of these primes is Marble's score. 2. Nora writes down a positive integer 3. Mable draws a vertical line between two adjacent primes she has written in step 1, and compute the product of the prime(s) on the left of the vertical line 4. Nora must add the product obtained by Marble in step 3 to the number she has written in step 2, and the sum becomes Nora's score. If Marble and Nora's scores have a common factor greater than 1, Marble wins, otherwise Nora wins. Who has a winning strategy?

2008 Postal Coaching, 2
Tags: prime number , divide , prime , number theory
Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

2019 Dürer Math Competition (First Round), P2
Tags: prime number , number theory
For a positive integer $n$ let $P(n)$ denote the set of primes $p$ for which there exist positive integers $a, b$ such that $n=a^p+b^p$ . Is it true that for any finite set $H$ consisting of primes, there is an n such that $P(n) = H$?

2015 Iran MO (3rd round), 3
Tags: number theory , prime number , quadratic residue
Let $p>5$ be a prime number and $A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}$ be the set of all quadratic residues modulo $p$, excluding zero. Prove that there doesn't exist any natural $a,c$ satisfying $(ac,p)=1$ such that set $B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}$ and set $A$ are disjoint modulo $p$. [i]This problem was proposed by Amir Hossein Pooya.[/i]

2017 Israel Oral Olympiad, 3
Tags: prime , number theory , prime number
2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

2015 Thailand TSTST, 1
Tags: prime number , number theory
Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.

2019 IOM, 1
Tags: prime number , number theory
Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

2005 China Western Mathematical Olympiad, 3
Tags: prime number , number theory , relatively prime
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2008 Bulgarian Autumn Math Competition, Problem 8.3
Tags: number theory , prime number , sum of digits
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.

2007 ITest, 1
Tags: prime number , number theory
A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.) $\textbf{(A) }4$

2019 Olympic Revenge, 2
Tags: number theory , perfect square , prime number , my 100th post
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$

2002 Korea Junior Math Olympiad, 2
Tags: prime number , number theory
Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.

2020 AMC 12/AHSME, 4
Tags: prime number
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

2007 Moldova Team Selection Test, 4
Tags: number theory , prime number
Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.

2024 Thailand TSTST, 4
Tags: number theory , prime number
The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

2016 Postal Coaching, 4
Tags: number theory , diophantine equation , prime number
Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

1973 USAMO, 5
Tags: 3d geometry , prime number , arithmetic sequence , number theory , geometry
Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

2024 Girls in Mathematics Tournament, 4
Tags: prime number , algebra , number theory
Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.

2016 IFYM, Sozopol, 2
Tags: prime number , algebra
Let $p$ be a prime number and the decimal notation of $\frac{1}{p}$ is periodical with a length of the period $4k$, $\frac{1}{p}=0,a_1 a_2…a_{4k} a_1 a_2…a_{4k}…$ .Prove that $a_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}$.

2016 IFYM, Sozopol, 2
Tags: polynomial , prime number , algebra , number theory
We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.