Olympiad problems

2012 Brazil Team Selection Test, 2

Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

1976 Spain Mathematical Olympiad, 4

Tags: number theory , divides

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

2023 Chile Junior Math Olympiad, 2

Tags: number theory , divides

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 $n!$.

1998 Tournament Of Towns, 1

Tags: number theory , divisible , divides

Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers? (Folklore)

1997 German National Olympiad, 6a

Tags: polynomial , primes , divides , algebra

Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

2000 Tournament Of Towns, 5

Tags: consecutive , number theory , divides , divisible

What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ? (S Tokarev)

1989 Tournament Of Towns, (217) 1

Tags: number theory , divides , divisible , Digits

Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

2015 Balkan MO Shortlist, N5

Tags: number theory , maximum , power of 2 , divides , Product , IMO Shortlist

For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

2019 Saudi Arabia IMO TST, 1

Tags: number theory , divides , divisible

Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$

2000 Estonia National Olympiad, 2

Tags: Pascal's Triangle , LCM , divides , number theory , divisible

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]

2013 Saudi Arabia BMO TST, 4

Tags: divides , divisible , number theory

Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.

2006 Singapore Senior Math Olympiad, 1

Tags: mutliple , divides , divisible , primes , number theory

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 $6$.

2017 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divides

Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that $m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.

2011 QEDMO 9th, 4

Tags: factorial , number theory , multiple , divides

Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

1996 Tournament Of Towns, (509) 2

Tags: number theory , divides , divisible

Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if (a) $d = 10$; (b) $d = 11$? (V Senderov)

2005 Estonia National Olympiad, 2

Tags: number theory , Sum of powers , divisible , divides

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

2008 Postal Coaching, 3

Tags: number theory , divides , divisible

Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.

2021 Saudi Arabia Training Tests, 34

Tags: polynomial , number theory , divides

Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

2022 Junior Balkan Team Selection Tests - Moldova, 11

Tags: divides , number theory

Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.

2018 Stanford Mathematics Tournament, 1

Tags: number theory , divides , divisible

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

1990 Tournament Of Towns, (265) 3

Tags: number theory , divides , Sum

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

2000 Singapore MO Open, 2

Tags: divides , divisible , primes , number theory , GCD

Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

2024 Regional Competition For Advanced Students, 4

Tags: number theory , divisible , divides

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]

2018 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , coprime , coprime numbers , Sum of powers , divides

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

2003 Bosnia and Herzegovina Junior BMO TST, 3

Tags: divides , divisible , number theory

Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.