Olympiad problems

1969 IMO Longlists, 30

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

2018 Ecuador Juniors, 6

Tags: number theory , composite numbers

What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?

PEN E Problems, 1

Tags: pen , composite numbers , number theory

Prove that the number $512^{3} +675^{3}+ 720^{3}$ is composite.

2015 Bosnia Herzegovina Team Selection Test, 3

Tags: composite numbers , number theory , infinitely many solutions , Divisibility

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

2014 IMAR Test, 2

Tags: composite numbers , number theory , Pell equations

Let $\epsilon$ be a positive real number. A positive integer will be called $\epsilon$-squarish if it is the product of two integers $a$ and $b$ such that $1 < a < b < (1 +\epsilon )a$. Prove that there are infinitely many occurrences of six consecutive $\epsilon$ -squarish integers.

2013 Costa Rica - Final Round, 2

Tags: number theory , odd , Even , Sum , composite numbers

Determine all even positive integers that can be written as the sum of odd composite positive integers.

2005 IMO Shortlist, 7

Tags: polynomial , number theory , composite numbers , algebra , IMO Shortlist

Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.

2022 Stars of Mathematics, 4

Tags: number theory , composite numbers , Fermat primes

Let $a{}$ be an even positive integer which is not a power of two. Prove that at least one of $2^{2^n}+1$ and $a^{2^n}+1$ is composite, for infinitely many positive integers $n$. [i]Bojan Bašić[/i]

2013 Hanoi Open Mathematics Competitions, 5

Tags: maximum , Summation , composite numbers

The number $n$ is called a composite number if it can be written in the form $n = a\times b$, where $a, b$ are positive integers greater than $1$. Write number $2013$ in a sum of $m$ composite numbers. What is the largest value of $m$? (A): $500$, (B): $501$, (C): $502$, (D): $503$, (E): None of the above.

2020/2021 Tournament of Towns, P1

Tags: number theory , composite numbers , Tournament of Towns

The number $2021 = 43 \cdot 47$ is composite. Prove that if we insert any number of digits “8” between 20 and 21 then the number remains composite. [i]Mikhail Evdikomov[/i]

1993 IMO Shortlist, 2

Tags: modular arithmetic , number theory , Divisibility , prime numbers , composite numbers , IMO Shortlist

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

1969 IMO Shortlist, 30

Tags: number theory , Sophie Germain identity , composite numbers , IMO Shortlist , IMO Longlist

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

2001 IMO Shortlist, 5

Tags: number theory , IMO , IMO 2001 , IMO Shortlist , composite numbers

Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.

2018 Romania Team Selection Tests, 2

Tags: number theory , quadratic reciprocity , Sum of Squares , composite numbers

Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.

1973 IMO Shortlist, 4

Tags: number theory , prime numbers , combinatorics , composite numbers , counting , IMO Shortlist

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?

2001 IMO, 6

Tags: number theory , IMO , IMO 2001 , IMO Shortlist , composite numbers

Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.