Olympiad problems

2014 Finnish National High School Mathematics, 4

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

2022 IMO Shortlist, N2

Tags: number theory , divsibility , primes

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , prime factorization , primes , consecutive

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

2021 239 Open Mathematical Olympiad, 1

Tags: primes , Integer , polynomial , Integer Polynomial , algebra

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

2019 Indonesia MO, 1

Tags: number theory , primes

Given that $n$ and $r$ are positive integers. Suppose that \[ 1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r) \] Prove that $n$ is a composite number.

1997 Austrian-Polish Competition, 5

Tags: Integer Polynomial , primes , polynomial , Cubic

Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

2013 Czech-Polish-Slovak Junior Match, 1

Tags: primes , number theory , infinitely many

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

2015 IFYM, Sozopol, 5

Tags: number theory , Divisibility , primes

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.

2018 Peru IMO TST, 10

Tags: recurrence relation , number theory , number theory with sequences , Product , primes

For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$. Define the sequence $a_1, a_2, a_3,...$ as followed: $a_1> 1$ is an arbitrary positive integer, $a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$. Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.

2023 Indonesia TST, 1

Tags: number theory , divsibility , primes

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2024 Abelkonkurransen Finale, 2a

Tags: algebra , algebra proposed , Sequence , primes

Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.

2002 IMO Shortlist, 3

Tags: algebra , number theory , primes , Divisors , IMO Shortlist

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2018 Saudi Arabia IMO TST, 1

Tags: algebra , Increasing , Sequence , primes , number theory

Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

2010 Estonia Team Selection Test, 1

Tags: number theory , GCD , primes , power of prime , coprime

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

2025 International Zhautykov Olympiad, 3

Tags: number theory , primes , construction

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

2015 Indonesia MO Shortlist, N1

Tags: number theory , Perfect Square , triplets , primes

A triple integer $(a, b, c)$ is called [i]brilliant [/i] when it satisfies: (i) $a> b> c$ are prime numbers (ii) $a = b + 2c$ (iii) $a + b + c$ is a perfect square number Find the minimum value of $abc$ if triple $(a, b, c)$ is [i]brilliant[/i].

2019 India PRMO, 21

Tags: number theory , Sets , primes , prime numbers

Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

2024 Regional Olympiad of Mexico Southeast, 1

Tags: primes , Mexico , integer pairs , prime divisor , number theory

Find all pairs of positive integers $a, b$ such that the numbers $a+1$, $b+1$, $2a+1$, $2b+1$, $a+3b$, and $b+3a$ are all prime numbers.

2012 Albania National Olympiad, 1

Tags: modular arithmetic , number theory , primes

Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.

2017 Balkan MO Shortlist, N5

Tags: number theory , arithmetic mean , fractional part , primes , odd , Quadratic Residues

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

2019 AMC 12/AHSME, 2

Tags: AMC , AMC 10 , AMC 10 B , 2019 AMC 10B , primes , AMC 12 B

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$