Olympiad problems

1979 Czech And Slovak Olympiad IIIA, 6

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

2020 Regional Olympiad of Mexico Northeast, 4

Tags: number theory , divide , prime , perfect square

Let $n > 1$ be an integer and $p$ be a prime. Prove that if $n|p-1$ and $p|n^3-1$, then $4p-3$ is a perfect square.

2003 Olympic Revenge, 5

Tags: prime number , prime , divisibility , number theory

Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

2009 VJIMC, Problem 2

Tags: prime , number theory

Prove that the number $$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.

2013 Greece JBMO TST, 3

Tags: prime , diophantine equation , number theory

If $p$ is a prime positive integer and $x,y$ are positive integers, find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1) If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).

2015 Chile National Olympiad, 2

Tags: prime , number theory

Find all prime numbers that do not have a multiple ending in $2015$.

2019 India PRMO, 14

Tags: prime , perfect square , relation

Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.

2013 Dutch BxMO/EGMO TST, 3

Tags: number theory , prime , diophantine equation

Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: prime , number theory

Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime

2014 Contests, 1

Tags: number theory , prime , factor , maximum

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

2014 Indonesia MO Shortlist, N1

Tags: number theory , prime , diophantine equation

(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number. (b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.

2022 AMC 10, 6

Tags: prime

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

2014 Finnish National High School Mathematics, 4

Tags: point , circles , prime , number theory

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 $r$.

2025 Israel TST, P2

Tags: prime , number theory

Prove that for all primes $ p $ such that $ p \equiv 3 \pmod{4} $ or $ p \equiv 5 \pmod{8} $, there exist integers \[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \] such that \[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]

2022 Serbia National Math Olympiad, P6

Tags: algebra , prime

Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$, divisible by $p$ or $q$.

1995 Korea National Olympiad, Day 2

Tags: greatest common divisor , algebra , integer polynomial , prime , factorization , polynomial

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

2017 Iran MO (3rd round), 1

Tags: divisibility , prime number , prime , number theory

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

2007 Korea Junior Math Olympiad, 8

Tags: prime , positive integer , divisor , number theory

Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infinite number of [i]Primes of the Year[/i].

2023 Indonesia TST, 1

Tags: number theory , prime , divisibility

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

2001 Estonia Team Selection Test, 5

Tags: prime , number theory , product , power , digit

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine , prime

Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , fraction , harmonic , number , prime divisor

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

2023 Germany Team Selection Test, 1

Tags: prime , divisibility , number theory

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

Oliforum Contest V 2017, 3

Tags: prime , number theory , product

Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$ (Paolo Leonetti)

2005 iTest, 17

Tags: number theory , sum of digits , prime

On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.