This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 24

2022 Stars of Mathematics, 4
Tags: composite number , fermat primes , number theory
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]

2014 Indonesia MO Shortlist, N2
Tags: composite number , composite , diophantine equation , number theory
Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.

2020 Greece National Olympiad, 4
Tags: composite number , composite , positive integer , number theory
Find all values of the positive integer $k$ that has the property: There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.

2014 IMAR Test, 2
Tags: pell equation , composite number , number theory
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.

1973 IMO Shortlist, 4
Tags: composite number , prime number , counting , number theory , combinatorics
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 ?

2011 Bundeswettbewerb Mathematik, 2
Tags: prime , composite , composite number , perfect square , number theory
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

2001 IMO Shortlist, 5
Tags: composite number , number theory
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.

2006 Germany Team Selection Test, 1
Tags: composite number , prime number , algebra , polynomial , number theory
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

PEN E Problems, 1
Tags: composite number , number theory
Prove that the number $512^{3} +675^{3}+ 720^{3}$ is composite.

2015 Bosnia Herzegovina Team Selection Test, 3
Tags: number theory , composite number , infinitely many solutions , divisibility
Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

2001 IMO, 6
Tags: composite number , number theory
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.

2013 Hanoi Open Mathematics Competitions, 5
Tags: composite number , maximum , summation
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.

1969 IMO Longlists, 30
Tags: composite number , sophie germain identity , number theory
$(GDR 2)^{IMO1}$ 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 $n.$

2005 IMO Shortlist, 7
Tags: polynomial , composite number , number theory , algebra
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.

2020/2021 Tournament of Towns, P1
Tags: number theory , composite number
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]

1969 IMO, 1
Tags: sophie germain identity , composite number , factorization , number theory
Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.

2018 Ecuador Juniors, 6
Tags: composite number , number theory
What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?

2005 IMO Shortlist, 3
Tags: algebra , polynomial , number theory , composite number , prime number
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

1993 IMO Shortlist, 2
Tags: number theory , divisibility , prime number , composite number , modular arithmetic
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 $P.$

2018 Romania Team Selection Tests, 2
Tags: number theory , quadratic reciprocity , sum of squares , composite number
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.

2022 Mexican Girls' Contest, 6
Tags: number theory , composite number , integer
Let $a$ and $b$ be positive integers such that $$\frac{5a^4+a^2}{b^4+3b^2+4}$$ is an integer. Prove that $a$ is not a prime number.

1969 IMO Shortlist, 30
Tags: number theory , sophie germain identity , composite number
$(GDR 2)^{IMO1}$ 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 $n.$

2013 Costa Rica - Final Round, 2
Tags: number theory , composite number , odd , even , sum
Determine all even positive integers that can be written as the sum of odd composite positive integers.

2006 Germany Team Selection Test, 1
Tags: polynomial , number theory , algebra , composite number , prime number
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.