Olympiad problems

2014 Indonesia MO Shortlist, N2

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.

2000 All-Russian Olympiad Regional Round, 10.1

Tags: composite , number theory

$2000$ numbers are considered: $11, 101, 1001, . . $. Prove that at least $99\%$ of these numbers are composite.

1997 Estonia National Olympiad, 1

Tags: prime , composite , number theory

Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.

2012 German National Olympiad, 1

Tags: number theory , sequence , recursive , composite

Define a sequence $(a_n)$ by $a_0 =-4 , a_1 =-7$ and $a_{n+2}= 5a_{n+1} -6a_n$ for $n\geq 0.$ Prove that there are infinitely many positive integers $n$ such that $a_n$ is composite.

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.

2020 Junior Macedonian National Olympiad, 1

Tags: set of positive integers , composite

Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$ there exist at least $k$ consecutive composite integers in the set {$n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$}.

2012 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: composite , number theory

Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$

2018 Danube Mathematical Competition, 1

Tags: divisor , composite , number theory

Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions: i) the number $n$ is composite; ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.

1987 All Soviet Union Mathematical Olympiad, 449

Tags: composite , sum , number theory , relatively prime

Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.

1977 Kurschak Competition, 1

Tags: number theory , composite , prime

Show that there are no integers $n$ such that $n^4 + 4^n$ is a prime greater than $5$.

2006 Bosnia and Herzegovina Junior BMO TST, 3

Tags: composite , number theory

Let $a, b, c, d$ be positive integers such that $ab = cd$. Prove that $w = a^{2006} + b^{2006} + c^{2006} + d^{2006}$ is composite.

2001 Estonia National Olympiad, 4

Tags: composite , sum of powers , prime , number theory

Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.

2013 Tournament of Towns, 2

Tags: number theory , digit , composite

Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?

2021 Harvard-MIT Mathematics Tournament., 3

Tags: number theory , composite

Let $m$ be a positive integer. Show that there exists a positive integer $n$ such that each of the $2m+1$ integers $$ 2^{n}-m,2^{n}-(m-1),\ldots,2^{n}+(m-1),2^{n}+m$$ is positive and composite.

1990 All Soviet Union Mathematical Olympiad, 531

Tags: composite , number theory

For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?

1985 Tournament Of Towns, (091) T2

Tags: combinatorics , number theory , difference , prime , composite , maximum

From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .

2018 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: prime , sequence , composite , infinitely many solutions , number theory

We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

1972 Czech and Slovak Olympiad III A, 4

Tags: number theory , composite , factoring

Show that there are infinitely many positive integers $a$ such that the number $n^4+a$ is composite for every positive integer $n.$ Give 5 (different) numbers $a$ with the mentioned property.

1999 Tournament Of Towns, 2

Tags: composite , odd , number theory

Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

1988 All Soviet Union Mathematical Olympiad, 475

Tags: composite , number theory

Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ .

1986 All Soviet Union Mathematical Olympiad, 418

Tags: algebra , polynomial , composite , trinomial

The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.

2010 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , composite , exponent

Prove that number $2^{2008}\cdot2^{2010}+5^{2012}$ is not prime