Olympiad problems

1997 Czech And Slovak Olympiad IIIA, 4

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

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 $x^3 + 3x + 14 = 2 p^n$

2007 Postal Coaching, 3

Tags: power of 3 , number theory , prime

Suppose $n$ is a natural number such that $4^n + 2^n + 1$ is a prime. Prove that $n = 3^k$ for some nonnegative integer $k$.

1978 IMO Shortlist, 17

Tags: quadratic , number theory , prime , sum of squares

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , prime number , prime , primitive root , decimal representation

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

2025 Nordic, 2

Tags: nt , number theory , prime

Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$ [size=75]$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.[/size]

2013 Balkan MO Shortlist, N5

Tags: number theory , diophantine equation , diophantine , prime

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , diophantine equation , prime number , prime

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2013 Tournament of Towns, 6

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

The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

VII Soros Olympiad 2000 - 01, 9.3

Tags: number theory , sum , prime

Write $102$ as the sum of the largest number of distinct primes.

1978 Bundeswettbewerb Mathematik, 4

Tags: number theory , prime , decimal representation , permutation

A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.

2014 Czech-Polish-Slovak Junior Match, 3

Tags: number theory , prime

Find with all integers $n$ when $|n^3 - 4n^2 + 3n - 35|$ and $|n^2 + 4n + 8|$ are prime numbers.

1969 Czech and Slovak Olympiad III A, 3

Tags: number theory , prime , fraction , sequence

Let $p$ be a prime. How many different (infinite) sequences $\left(a_k\right)_{k\ge0}$ exist such that for every positive integer $n$ \[\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?\]

2021 Indonesia TST, C

Tags: combinatorics , subset , counting , prime , summation

Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.

1999 Mexico National Olympiad, 2

Tags: arithmetic sequence , number theory , prime

Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.

2023 Indonesia TST, 1

Tags: number theory , prime , divisibility

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

2013 Thailand Mathematical Olympiad, 10

Tags: prime , perfect cube , number theory

Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.

2019 AMC 12/AHSME, 2

Tags: prime

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? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

2016 JBMO Shortlist, 2

Tags: combinatorics , sum , prime

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , prime

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

1994 Italy TST, 2

Tags: number theory , prime , perfect square

Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.

2019 Cono Sur Olympiad, 4

Tags: number theory , diophantine equation , prime

Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.

2010 NZMOC Camp Selection Problems, 3

Tags: number theory , prime

Find all positive integers n such that $n^5 + n + 1$ is prime.

2015 Latvia Baltic Way TST, 13

Tags: floor function , number theory , prime

Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ? $[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.

2010 Estonia Team Selection Test, 1

Tags: number theory , gcd , power of prime , coprime , prime

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.