Olympiad problems

1996 Singapore Team Selection Test, 3

Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$

2013 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide , divisible

Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns

2011 Belarus Team Selection Test, 1

Tags: number theory , odd , divisible , prime

Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with $a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$). Prove that a) $(a-1)\vdots p_i$ for some $i=1,..,n$ b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$? I. Bliznets

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 $169$.

2008 Hanoi Open Mathematics Competitions, 1

Tags: number theory , divisible , sum of digits

How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?

2020 Swedish Mathematical Competition, 5

Tags: number theory , combinatorics , divide , divisible

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$

1997 Tournament Of Towns, (524) 1

Tags: number theory , sum of digits , divisible

How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? (AI Galochkin)

2004 All-Russian Olympiad Regional Round, 9.5

Tags: combinatorics , number theory , divisible

The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?

2016 Saudi Arabia BMO TST, 3

Tags: number theory , divide , divisible

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

1955 Moscow Mathematical Olympiad, 290

Tags: divide , number theory , divisible

Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

2003 Estonia Team Selection Test, 2

Tags: number theory , divisible

Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$. (H. Nestra)

2013 Flanders Math Olympiad, 1

Tags: number theory , digit , divisible , sum

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

2007 Thailand Mathematical Olympiad, 11

Tags: number theory , combinatorics , divisible , divide

Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.

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 $d^2 + n^2$.

2003 Austrian-Polish Competition, 7

Tags: number theory , divide , factorial , divisible

Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $. Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

2010 Saudi Arabia Pre-TST, 1.3

Tags: number theory , divide , digit , divisible

1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.

1969 Swedish Mathematical Competition, 5

Tags: number theory , divide , divisible

Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.

2012 Switzerland - Final Round, 7

Tags: number theory , divide , divisible

Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.

2008 Abels Math Contest (Norwegian MO) Final, 1

Tags: number theory , divisible , integer

Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$. (a) Show that $s(n)$ is an integer whenever $n$ is an integer. (b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?

1998 Tournament Of Towns, 3

Tags: combinatorics , divisible , divide

Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.) (G Galperin)

2014 Saudi Arabia Pre-TST, 2.1

Tags: divide , divisible , number theory

Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.

2014 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , digit , divisible

Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

2017 Czech-Polish-Slovak Junior Match, 3

Tags: digit , divisible , number theory

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

1999 Kazakhstan National Olympiad, 6

Tags: number theory with sequences , sequence , divisible , remainder , number theory

In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

2001 Paraguay Mathematical Olympiad, 3

Tags: number theory , sum of digits , divisible

Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.