Olympiad problems

2005 USAMO, 1

Tags: induction , number theory , prime numbers , Arrangements , relatively prime , Divisors , combinatorics

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

1987 IMO Longlists, 69

Tags: number theory , polynomial , prime numbers , quadratics , IMO Shortlist , IMO , IMO 1987

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

2013 Korea - Final Round, 5

Tags: modular arithmetic , Euler , number theory , prime numbers , number theory proposed

Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties \[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \] Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.

2016 China Team Selection Test, 4

Tags: number theory , Sequence , prime numbers

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

2012 Brazil Team Selection Test, 1

Tags: algorithm , number theory , prime numbers , Divisibility , IMO Shortlist , number of divisors

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

1997 Korea National Olympiad, 4

Tags: number theory , prime numbers

For any prime number $p>2,$ and an integer $a$ and $b,$ if $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{(p-1)^3}=\frac{a}{b},$ prove that $a$ is divisible by $p.$

2011 Pre-Preparation Course Examination, 5

Tags: calculus , integration , logarithms , inequalities , function , number theory , prime numbers

suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent: [b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$ [b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).

2003 Olympic Revenge, 5

Tags: number theory , prime numbers , prime , Divisibility

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 $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

2013 Dutch IMO TST, 3

Tags: number theory , prime numbers , number theory unsolved

Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

2023 Brazil EGMO Team Selection Test, 2

Tags: number theory , Sets , prime numbers

Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions: $(i)$ the prime factors of any element of $B$ are in $A$; $(ii)$ no term of $B$ divides another element of this set.

2012 Middle European Mathematical Olympiad, 4

Tags: number theory , prime numbers , number theory proposed

The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and \[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \] for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $.

2004 Baltic Way, 8

Tags: algebra , polynomial , inequalities , function , logarithms , number theory , prime numbers

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

2008 Czech-Polish-Slovak Match, 3

Tags: modular arithmetic , quadratics , number theory , prime numbers , number theory proposed

Find all primes $p$ such that the expression \[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\] is divisible by $p^3$.

2012 Morocco TST, 1

Tags: number theory unsolved , number theory , prime numbers

Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

2018 Dutch BxMO TST, 3

Tags: number theory , prime numbers , Divisibility

Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

2013 China Girls Math Olympiad, 4

Tags: algebra , polynomial , function , modular arithmetic , number theory , prime numbers

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

2017 India PRMO, 28

Tags: number theory , prime numbers , Fermat Pseudo-Primes , PRMO , Euler s Theorem , carmichael s number , Hi

Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.

2022 Korea National Olympiad, 8

Tags: number theory , prime numbers

$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation. $$p^2 x^4-6px^2+1=y^2$$

2017 Bundeswettbewerb Mathematik, 4

Tags: number theory , number theory unsolved , Sequence , recursion , prime numbers , Integer

The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$: (a) The number $a_n$ is a positive integer. (b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$. (c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.

2013 Peru IMO TST, 5

Tags: number theory , prime numbers , binomial coefficients , IMO Shortlist

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2020 Kosovo National Mathematical Olympiad, 3

Tags: number theory , prime numbers , national olympiad , OLYM

Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.

2008 Tuymaada Olympiad, 2

Tags: AMC , USA(J)MO , USAMO , number theory , prime numbers , number theory unsolved

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]