Olympiad problems

1989 Poland - Second Round, 4

The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.

2022 Regional Olympiad of Mexico West, 5

Tags: number theory , floor function , divide

Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.

2012 China Northern MO, 3

Tags: number theory , divide

Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

1993 Swedish Mathematical Competition, 1

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

An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.

1979 Chisinau City MO, 174

Tags: number theory , odd , divide , divisible

Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2018 Costa Rica - Final Round, 5

Tags: number theory , divisible , divide

Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

2017 Abels Math Contest (Norwegian MO) Final, 2

Tags: number theory , sequence , divide

Let the sequence an be defined by $a_0 = 2, a_1 = 15$, and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$. Show that there are infinitely many integers $k$ such that $269 | a_k$.

1999 Ukraine Team Selection Test, 8

Tags: number theory , divide

Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.

2014 Switzerland - Final Round, 5

Tags: number theory , divisible , divide

Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have: $$\sum_{d | n} a_d = 2^n.$$ Show for every $n \in N$ that $n$ divides $a_n$. Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$

1997 Chile National Olympiad, 2

Tags: number theory , divisible , divide

Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$. Prove that $a \equiv n \equiv 1 \mod 10$ .

2016 Czech-Polish-Slovak Junior Match, 2

Tags: divide , digit , number theory

Find the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits. Czech Republic

2016 Czech-Polish-Slovak Junior Match, 3

Tags: geometry , combinatorial geometry , combinatorics , divide , 3d geometry , prism

Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

1997 Singapore MO Open, 2

Tags: number theory , binomial , divide , divisible

Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

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 $(n^n)!$ .

1990 Greece National Olympiad, 3

Tags: number theory , divide , divisible

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

2015 Dutch IMO TST, 5

Tags: number theory , positive integer , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2016 Saudi Arabia Pre-TST, 1.3

Tags: number theory , odd , divide

Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.

2024 Regional Competition For Advanced Students, 4

Tags: number theory , divisible , divide

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

2021 Polish Junior MO Second Round, 3

Tags: number theory , divisible , divide

Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.

2014 Czech-Polish-Slovak Junior Match, 5

Tags: game , number theory , divide , divisor

There is the number $1$ on the board at the beginning. If the number $a$ is written on the board, then we can also write a natural number $b$ such that $a + b + 1$ is a divisor of $a^2 + b^2 + 1$. Can any positive integer appear on the board after a certain time? Justify your answer.

2023 Chile National Olympiad, 1

Tags: number theory , divide

Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.

1995 Singapore MO Open, 4

Tags: number theory , divisible , divide

Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

2022 Regional Olympiad of Mexico West, 1

Tags: number theory , divide

Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.

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.

2016 JBMO Shortlist, 2

Tags: number theory , divide

Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions: a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.