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

2015 Switzerland - Final Round, 9
Tags: divide , divisible , number theory
Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

2015 Balkan MO Shortlist, N3
Tags: prime divisor , integer , divide , divisor , number theory
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$. Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer. (FYROM)

2012 NZMOC Camp Selection Problems, 4
Tags: divide , multiple , divisible , number theory , prime
A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

1993 All-Russian Olympiad Regional Round, 9.2
Tags: divide , divisible , number theory
Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

2011 Saudi Arabia Pre-TST, 3.1
Tags: divide , divisible , number theory
Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

2007 Estonia National Olympiad, 1
Tags: divide , digit , number theory
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

2015 Gulf Math Olympiad, 1
Tags: divide , divisible , odd , prime , number theory
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.

2006 Thailand Mathematical Olympiad, 3
Tags: divide , algebra , polynomial
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 $P(x) - Q(x)$.

VMEO III 2006 Shortlist, N6
Tags: divide , number theory
Find all sets of natural numbers $(a, b, c)$ such that $$a+1|b^2+c^2\,\, , b+1|c^2+a^2\,\,, c+1|a^2+b^2.$$

2005 Thailand Mathematical Olympiad, 11
Tags: divide , number theory
Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.

1947 Kurschak Competition, 1
Tags: divide , divisible , number theory
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

1969 Swedish Mathematical Competition, 5
Tags: divide , divisible , number theory
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 $3$.

2005 Estonia National Olympiad, 2
Tags: sum of powers , divide , number theory , divisible
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 $7$.

1910 Eotvos Mathematical Competition, 2
Tags: divide , multiple , number theory
Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2000 All-Russian Olympiad Regional Round, 9.2
Tags: divide , divisible , number theory
Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?

2012 Austria Beginners' Competition, 1
Tags: divide , divisible , number theory
Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$. Prove that the product $a\cdot b$ is always divisible by $14$.

2017 India PRMO, 1
Tags: sum of digits , divide , number theory , divisible
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2000 Tournament Of Towns, 2
Tags: divide , divisible , number theory
Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$. (A Spivak)

2023 Francophone Mathematical Olympiad, 4
Tags: divide , product , number theory , consecutive
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 $b(b+1)\cdots(b+2023)$?

2018 Singapore Junior Math Olympiad, 1
Tags: divide , digit , multiple , number theory
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 $37$.

1999 Denmark MO - Mohr Contest, 5
Tags: digit , divide , divisible , number theory
Is there a number whose digits are only $1$'s and which is divided by $1999$?

2011 Indonesia TST, 4
Tags: sum of divisors , divide , prime , number theory
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2007 Switzerland - Final Round, 9
Tags: integer , divide , number theory , divisible
Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.

2006 Switzerland - Final Round, 10
Tags: divide , divisible , number theory
Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

1963 German National Olympiad, 1
Tags: divide , remainder , number theory
a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!