Olympiad problems

2017 India PRMO, 1

How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2003 Austria Beginners' Competition, 3

Tags: number theory , consecutive , divide , divisible

a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

Mathley 2014-15, 8

Tags: divisible , power of 2 , divide , number theory

For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$ where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$. Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi

1998 Singapore Senior Math Olympiad, 1

Tags: integer , divisible , divide , number theory

Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

1990 Tournament Of Towns, (265) 3

Tags: number theory , divide , sum

Find $10$ different positive integers such that each of them is a divisor of their sum (S Fomin, Leningrad)

2013 Saudi Arabia GMO TST, 3

Tags: divide , divisible , number theory

Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

1949-56 Chisinau City MO, 7

Tags: factorial , number theory , divisible , prime , divide

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2015 Dutch BxMO/EGMO TST, 1

Tags: number theory , divisor , divide

Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2021 Dutch IMO TST, 3

Tags: number theory , divide , divisible

Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

1988 Tournament Of Towns, (186) 3

Tags: number theory , divisible , sum , divide

Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

2014 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , divisible , divide , product

Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

2006 Singapore Senior Math Olympiad, 1

Tags: mutliple , divide , divisible , number theory , prime

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2003 Austrian-Polish Competition, 4

Tags: divide , divisible , product , number theory

A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.

1947 Kurschak Competition, 1

Tags: number theory , divide , divisible

Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

2017 Switzerland - Final Round, 4

Tags: number theory , divide , divisible

Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$ $$n + 2 | n^p + q^p.$$ Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.

2008 Mathcenter Contest, 6

Tags: number theory , divide

For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$. [i](tomoyo-jung)[/i]

2014 Chile National Olympiad, 4

Tags: number theory , divide , divisible

Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.

1971 Spain Mathematical Olympiad, 8

Tags: number theory , divide

Among the $2n$ numbers $1, 2, 3, . . . , 2n$ are chosen in any way $n + 1$ different numbers. Prove that among the chosen numbers there are at least two, such that one divides the other.

2010 Saudi Arabia BMO TST, 3

Tags: number theory , divisible , divide

How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

1910 Eotvos Mathematical Competition, 2

Tags: number theory , multiple , divide

Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2015 NZMOC Camp Selection Problems, 8

Tags: divide , divisible , divisor , number theory

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.

1996 Estonia National Olympiad, 4

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

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

2013 Saudi Arabia Pre-TST, 4.2

Tags: number theory , divide , divisible

Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

1986 Tournament Of Towns, (108) 2

Tags: number theory , maximum , digit , divide

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

2011 Indonesia TST, 4

Tags: number theory , prime , sum of divisors , divide

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.