Olympiad problems

2016 Croatia Team Selection Test, Problem 4

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

2011 Silk Road, 4

Tags: number theory , representation , prime

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

2012 Tournament of Towns, 2

Tags: prime , combinatorics , table , rectangle table

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

2018 Chile National Olympiad, 1

Tags: number theory , prime number , prime

Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

2021 Nigerian MO Round 3, Problem 1

Tags: number theory , diophantine equation , prime

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

2015 India IMO Training Camp, 2

Tags: number theory , prime

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

1992 Romania Team Selection Test, 6

Tags: number theory , prime , integer polynomial , prime number

Let $m,n$ be positive integers and $p$ be a prime number. Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.

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.

1986 Polish MO Finals, 3

Tags: number theory , sum , divisible , prime , prime divisor

$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.

2008 Singapore Junior Math Olympiad, 4

Tags: sum , number theory , prime

Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?

2018 Peru IMO TST, 10

Tags: recurrence relation , number theory , number theory with sequences , product , prime

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.

Oliforum Contest V 2017, 4

Tags: number theory , prime

Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and define $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

2015 Regional Olympiad of Mexico Southeast, 1

Tags: number theory , prime

Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.

2024 SG Originals, Q5

Tags: combinatorics , grid , number theory , prime number , prime

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2014 Ukraine Team Selection Test, 12

Tags: prime , number theory , divisibility

Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.

2015 Brazil Team Selection Test, 2

Tags: number theory , prime

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

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 QEDMO 13th or 12th, 6

Tags: number theory , prime

A composite natural number $n$ is called [i]happy [/i] if at most one of the numbers $2^{2^n}+ 1$ and $6^{2^n}+ 1$ is prime. Show that there are infinitely many happy numbers.

2009 Korea Junior Math Olympiad, 1

Tags: number theory , multiple , product , prime

For primes $a, b,c$ that satisfy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

2013 India Regional Mathematical Olympiad, 2

Tags: divisor , prime

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

2021 Romanian Master of Mathematics Shortlist, N2

Tags: number theory , combinatorial number theory , prime

We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.

2021 Durer Math Competition Finals, 1

Tags: number theory , prime

Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.

2023 USAMO, 5

Tags: arithmetic sequence , modular arithmetic , prime

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

1996 Greece Junior Math Olympiad, 4b

Tags: diophantine equation , diophantine , prime , number theory

Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.

2018 Malaysia National Olympiad, A6

Tags: number theory , prime

A [i]semiprime [/i] is a positive integer that is a product of two prime numbers. For example, $9$ and $10$ are semiprimes. How many semiprimes less than $100$ are there?