Olympiad problems

2000 Portugal MO, 3

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2020 Regional Olympiad of Mexico Northeast, 4

Tags: number theory , divide , prime , perfect square

Let $n > 1$ be an integer and $p$ be a prime. Prove that if $n|p-1$ and $p|n^3-1$, then $4p-3$ is a perfect square.

2016 India Regional Mathematical Olympiad, 3

Tags: number theory , divide , divisor

$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$

2006 Thailand Mathematical Olympiad, 3

Tags: algebra , polynomial , divide

Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$. Show that $x - 1$ divides $P(x) - Q(x)$.

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 $24$.

2013 Thailand Mathematical Olympiad, 1

Tags: divide , prime , number theory

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

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.

2013 Korea Junior Math Olympiad, 4

Tags: number theory , prime number , minimum , divide

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

2021 Auckland Mathematical Olympiad, 4

Tags: number theory , divide , divisible , power

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

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 $\ge 3$.

2023 Chile Junior Math Olympiad, 2

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 $n!$.

2014 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , divide , divisible , digit

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: divide , number theory

Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

2013 Saudi Arabia GMO TST, 4

Tags: number theory , divide , divisible

Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.

1994 Poland - Second Round, 6

Tags: number theory , divide , prime

Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.

1983 Austrian-Polish Competition, 8

Tags: number theory , recurrence relation , sequence , divide , factorial , divisible , product

(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

2014 Switzerland - Final Round, 5

Tags: number theory , divide , divisible

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 $a_1 + a_2 + a_3 + a_6 = 2^6.$

2000 Chile National Olympiad, 3

Tags: number theory , divide , divisible , periodic

A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$

2010 Thailand Mathematical Olympiad, 3

Tags: number theory , divisible , divide

Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

2018 Singapore Junior Math Olympiad, 1

Tags: multiple , number theory , divide , digit

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

2023 Francophone Mathematical Olympiad, 4

Tags: number theory , divide , consecutive , product

Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?

1955 Moscow Mathematical Olympiad, 290

Tags: number theory , divisible , divide

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

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 $a = b$.

2019 Auckland Mathematical Olympiad, 2

Tags: number theory , divisible , divide

Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.

2016 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , divide , digit

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