This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 387

2009 Kyiv Mathematical Festival, 5
Tags: number theory , number theory with sequences , divide , sequence
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?

2015 Saudi Arabia Pre-TST, 3.3
Tags: number theory , divide
Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$. (Malik Talbi)

1982 Poland - Second Round, 5
Tags: number theory , divide , divisible
Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

1989 Tournament Of Towns, (205) 3
Tags: number theory , divisible , divide , digit
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

1911 Eotvos Mathematical Competition, 3
Tags: number theory , divide , divisible
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.

2019 Saudi Arabia BMO TST, 1
Tags: number theory , divide , divisible , prime
Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

2022 China Northern MO, 2
Tags: divide , number theory
(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

2023 Assara - South Russian Girl's MO, 2
Tags: number theory , divisible , divide
The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

2006 May Olympiad, 1
Tags: number theory , divide
Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.

2021 Saudi Arabia BMO TST, 3
Tags: number theory , divide , divisible
Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

2012 India Regional Mathematical Olympiad, 2
Tags: number theory , divisibility , divide , divisor
Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

2018 NZMOC Camp Selection Problems, 1
Tags: multiple , divide , number theory
Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.

1998 Tournament Of Towns, 1
Tags: number theory , divisible , divide
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)

2013 Czech-Polish-Slovak Junior Match, 4
Tags: divide , divisor , number theory , digit
Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

2012 India Regional Mathematical Olympiad, 2
Tags: number theory , divide , divisible , power
Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

2016 Dutch BxMO TST, 5
Tags: number theory , divide
Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

2017 Balkan MO Shortlist, N3
Tags: number theory , divide , exponential
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.

1954 Moscow Mathematical Olympiad, 267
Tags: divide , polynomial , divisible , algebra
Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

2017 May Olympiad, 5
Tags: number theory , divide , divisible
We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.

2019 Austrian Junior Regional Competition, 4
Tags: number theory , divisible , divide , prime
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

2002 Portugal MO, 4
Tags: number theory , divide , digit
The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

1978 Chisinau City MO, 159
Tags: number theory , prime , divide , divisible , factorial
Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

2010 Saudi Arabia IMO TST, 3
Tags: recurrence relation , divide , number theory
Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$. Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .

2005 Thailand Mathematical Olympiad, 12
Tags: number theory , divide , divisible , even
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.

2011 Saudi Arabia IMO TST, 3
Tags: algebra , polynomial , number theory , divide
Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.