Olympiad problems

2012 Singapore Junior Math Olympiad, 5

Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number. (Note: Two positive integers $m, n$ are coprime if their only common factor is 1)

2016 Thailand Mathematical Olympiad, 6

Tags: prime , power of 2 , number theory

Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$.

2019 Centers of Excellency of Suceava, 2

Tags: prime , number theory , gcd

Let $ \left( s_n \right)_{n\ge 1 } $ be a sequence with $ s_1 $ and defined recursively as $ s_{n+1}=s_n^2-s_n+1. $ Prove that any two terms of this sequence are coprime. [i]Dan Nedeianu[/i]

1997 German National Olympiad, 6a

Tags: polynomial , divide , algebra , prime

Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

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.

2023 Indonesia TST, 1

Tags: number theory , divisibility , prime

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

2003 IMO Shortlist, 7

Tags: modular arithmetic , polynomial , recurrence , sequence , divisibility , prime

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

2006 Thailand Mathematical Olympiad, 11

Tags: number theory , remainder , product , prime

Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

2022 Bulgarian Spring Math Competition, Problem 10.4

Tags: algebra , system of equations , prime

Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy \[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]

2021 Indonesia MO, 3

Tags: number theory , sequence , prime

A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

2017 Balkan MO Shortlist, N5

Tags: quadratic residue , number theory , arithmetic mean , fractional part , odd , prime

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 $p$ .

2024 Mexican Girls' Contest, 8

Tags: prime , number theory

Find all positive integers $n$ such that among the $n$ numbers \[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \] there are $n$, $n - 1$, or $n - 2$ primes.

2013 Hanoi Open Mathematics Competitions, 1

Tags: minimum , number theory , prime

Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is: (A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.

2006 Junior Tuymaada Olympiad, 2

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

Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

1993 Spain Mathematical Olympiad, 4

Tags: number theory , prime , multiple

Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.

2021 New Zealand MO, 3

Tags: number theory , prime

Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?

2024 Singapore MO Open, 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]

2009 Postal Coaching, 2

Tags: fraction , prime , number theory

Find all pairs $(x, y)$ of natural numbers $x$ and $y$ such that $\frac{xy^2}{x+y}$ is a prime

2025 Bulgarian Spring Mathematical Competition, 9.4

Tags: functional equation , number theory , divisibility , sophie germain identity , function , prime

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.

2012 Mathcenter Contest + Longlist, 3

Tags: divisible , prime , number theory

If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

2019 India PRMO, 21

Tags: number theory , prime number , set , prime

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 $N$.