Olympiad problems

2012 NZMOC Camp Selection Problems, 4

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2007 Austria Beginners' Competition, 1

Tags: number theory , divide , divisible

Prove that the number $9^n+8^n+7^n+6^n-4^n-3^n-2^n-1^n$ is divisible by $10$ for all non-negative $n$.

2013 Balkan MO Shortlist, N4

Tags: number theory , divisible , sum of powers , divisor

Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.

2004 Thailand Mathematical Olympiad, 15

Tags: divide , divisible , number theory

Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.

1992 All Soviet Union Mathematical Olympiad, 562

Tags: number theory , divisible , 4-digit

Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?

2017 Latvia Baltic Way TST, 6

Tags: divisible , number theory , combinatorics

A natural number is written in each box of the $13 \times 13$ grid area. Prove that you can choose $2$ rows and $4$ columns such that the sum of the numbers written at their $8$ intersections is divisible by $8$.

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divide , divisible , power

Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

1976 Chisinau City MO, 126

Tags: algebra , polynomial , divisible

Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

1975 Chisinau City MO, 87

Tags: number theory , divisible , difference

Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.

2020 Kosovo National Mathematical Olympiad, 2

Tags: number theory , divisible

Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.

2022 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , divisible

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

2015 Puerto Rico Team Selection Test, 8

Tags: multiple , divisible , divide , number theory

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

2006 Austria Beginners' Competition, 1

Tags: number theory , divisible

Do integers $a, b$ exist such that $a^{2006} + b^{2006} + 1$ is divisible by $2006^2$?

1990 ITAMO, 5

Tags: number theory , divisible , trinomial

Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.

2005 Estonia National Olympiad, 2

Tags: divisible , divide , sum of powers , number theory

Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.

2019 Durer Math Competition Finals, 6

Tags: number theory , multiple , divide , divisible , digit

Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

1989 Tournament Of Towns, (210) 4

Tags: number theory , divide , divisible

Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .

2015 Caucasus Mathematical Olympiad, 2

Tags: number theory , divisible , maximum

There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?

2001 Austria Beginners' Competition, 1

Tags: number theory , divide , divisible

Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

2011 Belarus Team Selection Test, 1

Tags: combinatorics , number theory , multiple , divisible

Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ? I. Gorodnin

2002 Singapore Team Selection Test, 2

Tags: floor function , number theory , integer , divide , divisible

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

2021 Auckland Mathematical Olympiad, 4

Tags: number theory , divide , divisible , power

Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

1953 Moscow Mathematical Olympiad, 234

Tags: number theory , divisible , minimum

Find the smallest number of the form $1...1$ in its decimal expression which is divisible by $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.

2025 Romania EGMO TST, P2

Tags: divisible , number theory , inequalities , subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

2008 Switzerland - Final Round, 3

Tags: number theory , divide , divisible

Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.