Olympiad problems

2016 IFYM, Sozopol, 2

We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.

2013 JBMO TST - Turkey, 2

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

[b]a)[/b] Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares. [b]b)[/b] Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?

2009 Germany Team Selection Test, 2

Tags: induction , number theory , prime numbers , number theory unsolved

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

2015 Taiwan TST Round 2, 2

Tags: IMO Shortlist , combinatorics , algebra , prime numbers

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

2020 Hong Kong TST, 4

Tags: number theory , prime numbers , floor function

Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

1973 USAMO, 5

Tags: geometry , 3D geometry , number theory , prime numbers , arithmetic sequence

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

2015 Turkey Junior National Olympiad, 3

Tags: number theory , Diophantine equation , prime numbers

Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

2015 Israel National Olympiad, 7

Tags: Fibonacci , Fibonacci sequence , prime numbers , Divisibility , number theory , combinatorics

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

2021 Kyiv City MO Round 1, 8.5

Tags: number theory , prime numbers

For a prime number $p > 3$, define the following irreducible fraction: $$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$ Prove that $m$ is divisible by $p$. [i]Proposed by Oleksii Masalitin[/i]

2011 IMO Shortlist, 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]

2008 Romania National Olympiad, 3

Tags: inequalities , number theory , prime numbers

Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.

2002 AMC 10, 14

Tags: quadratics , Vieta , algebra , number theory , prime numbers , special factorizations

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

1997 Romania Team Selection Test, 4

Tags: floor function , ceiling function , number theory , prime numbers , combinatorics proposed , combinatorics , poset

Let $p,q,r$ be distinct prime numbers and let \[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \] Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$. [i]Ioan Tomescu[/i]

2016 India IMO Training Camp, 3

Tags: number theory , set , prime numbers

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

2003 AMC 10, 14

Tags: number theory , prime numbers

Let $ n$ be the largest integer that is the product of exactly $ 3$ distinct prime numbers, $ d$, $ e$, and $ 10d\plus{}e$, where $ d$ and $ e$ are single digits. What is the sum of the digits of $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

2009 Iran Team Selection Test, 2

Tags: pigeonhole principle , modular arithmetic , inequalities , number theory , prime numbers , Diophantine equation , number theory proposed

Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

2023 Argentina National Olympiad, 4

Tags: number theory , prime numbers

Lets say that a positive integer is $good$ if its equal to the the subtraction of two positive integer cubes. For example: $7$ is a $good$ prime because $2^3-1^3=7$. Determine how much the last digit of a $good$ prime may be worth. Give all the possibilities.

2020 IMC, 6

Tags: IMC , number theory , prime numbers , IMC 2020

Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$

2014 Junior Regional Olympiad - FBH, 4

Tags: number theory , prime numbers

Find all prime numbers $p$ and $q$ such that $$(2p-q)^2=17p-10q$$

2017 Argentina National Math Olympiad Level 2, 4

Tags: number theory , prime numbers

Find all positive integers $a$ such that $4x^2 + a$ is prime for all $x = 0, 1, \dots, a - 1$.

2023 Durer Math Competition Finals, 1

Tags: number theory , algebra , prime numbers

Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

2020 New Zealand MO, 4

Tags: primes , number theory , prime numbers

Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.

1987 IMO Shortlist, 20

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]

STEMS 2021 Math Cat B, Q2

Tags: polynomial , power of 2 , prime numbers , Divisibility , kobayashi

Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

2020 Poland - Second Round, 5.

Tags: number theory , prime numbers

Let $p>$ be a prime number and $S$ be a set of $p+1$ integers. Prove that there exist pairwise distinct numbers $a_1,a_2,...,a_{p-1}\in S$ that $$ a_1+2a_2+3a_3+...+(p-1)a_{p-1}$$ is divisible by $p$.