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

2010 Bundeswettbewerb Mathematik, 1
Tags: number theory , prime numbers , Digits
Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?

2024 Romanian Master of Mathematics, 2
Tags: algorithm , number theory , prime numbers , Partial sums , permutations , RMM
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]

2010 Bosnia Herzegovina Team Selection Test, 1
Tags: number theory , prime numbers , number theory proposed
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

1999 Singapore Team Selection Test, 3
Tags: number theory , prime numbers , Divide
Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$

1994 IMO, 6
Tags: modular arithmetic , number theory , prime numbers , Combinatorial Number Theory , IMO , IMO 1994 , combinatorics
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

1988 IMO Longlists, 94
Tags: number theory , prime numbers , Perfect Square , IMO Longlist
Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.

2019 Dürer Math Competition (First Round), P2
Tags: number theory , prime numbers
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 $P(n) = H$?

2018 USAMTS Problems, 5:
Tags: number theory , prime numbers , usamts Year 30
The sequence $\{a_n\}$ is defined by $a_0 = 1, a_1 = 2,$ and for $n \geq 2,$ $$a_n = a_{n-1}^2 + (a_0a_1 \dots a_{n-2})^2.$$ Let $k$ be a positive integer, and let $p$ be a prime factor of $a_k.$ Show that $p > 4(k-1).$

2016 IFYM, Sozopol, 2
Tags: algebra , prime numbers
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 $a_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}$.

2010 Contests, 3
Tags: inequalities , number theory , prime numbers , number theory proposed
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

2016 Latvia National Olympiad, 4
Tags: number theory , prime numbers
In a Pythagorean triangle all sides are longer than 5. Is it possible that (a) all three sides are prime numbers, (b) exactly two sides are prime numbers. (Note: We call a triangle "Pythagorean", if it is a right-angled triangle where all sides are positive integers.)

2022 IMO, 3
Tags: number theory , prime numbers , IMO , IMO 2022
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

2008 Romania Team Selection Test, 2
Tags: number theory , prime numbers , number theory proposed
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?

2015 Germany Team Selection Test, 3
Tags: IMO Shortlist , combinatorics , algebra , prime numbers
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

2007 Moldova Team Selection Test, 4
Tags: number theory , prime numbers , number theory proposed
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 $p|n!+1$.

1977 IMO Longlists, 27
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.)

2020 Junior Balkаn MO, 4
Tags: number theory , prime numbers , Junior Balkan , Junior , jbmo2020 , geometry
Find all prime numbers $p$ and $q$ such that $$1 + \frac{p^q - q^p}{p + q}$$ is a prime number. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2024 CIIM, 3
Tags: number theory , euler totient function , prime numbers , Summation
Given a positive integer \(n\), let \(\phi(n)\) denote the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). Find all possible positive integers \(k\) for which there exist positive integers \(1 \leq a_1 < a_2 < \dots < a_k\) such that: \[ \left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024 \]

2019 IMEO, 4
Tags: polynomial , prime numbers , algebra , algebra proposed
Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well. [i]Proposed by Valentio Iverson (Indonesia)[/i]

1993 Irish Math Olympiad, 2
Tags: number theory , prime numbers , number theory unsolved
A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.

1976 Chisinau City MO, 122
Tags: Cyclic , geometry , areas , number theory , prime numbers
The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

2014 Contests, 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 $f(f(n)) = 2$.

2017 Macedonia JBMO TST, 1
Tags: number theory , prime numbers , Divisibility
Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

2010 All-Russian Olympiad, 3
Tags: logarithms , number theory , prime numbers , number theory proposed
Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.

2018 Malaysia National Olympiad, A6
Tags: number theory , powers , prime numbers
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.