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: 367

2024 Austrian MO Regional Competition, 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]

1996 Austrian-Polish Competition, 1
Tags: number theory , divides , divisible , Digits
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

2021 Czech and Slovak Olympiad III A, 4
Tags: number theory , divisor , divides
Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$). (Tomáš Bárta)

2005 Estonia National Olympiad, 2
Tags: divisible , divides , number theory , Sum of powers
Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.

2014 Saudi Arabia Pre-TST, 4.1
Tags: number theory , inequalities , divides , divisible
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

2000 Chile National Olympiad, 5
Tags: power of 2 , divides , divisible , number theory
Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.

2013 Saudi Arabia GMO TST, 4
Tags: number theory , divides , divisible
Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.

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

2020 Colombia National Olympiad, 4
Tags: arithmetic sequence , divides , number theory
Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.

2011 Saudi Arabia Pre-TST, 4.4
Tags: number theory , divisor , divides
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.

2022 Regional Olympiad of Mexico West, 1
Tags: number theory , divides
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 $b$.

2012 Austria Beginners' Competition, 1
Tags: number theory , divides , divisible
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 $14$.

2016 Dutch Mathematical Olympiad, 3
Tags: greatest common divisor , GCD , number theory , divisor , divides
Find all possible triples $(a, b, c)$ of positive integers with the following properties: • $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$, • $a$ is a divisor of $a + b + c$, • $b$ is a divisor of $a + b + c$, • $c$ is a divisor of $a + b + c$. (Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)

2019 Ecuador NMO (OMEC), 3
Tags: divisible , divides , power of 2 , number theory
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

2009 Cuba MO, 9
Tags: number theory , divides
Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$

2013 QEDMO 13th or 12th, 2
Tags: Binomial , number theory , divisible , divides
Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

2022 New Zealand MO, 5
Tags: number theory , multiple , divides , divisible , recurrence relation
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

1995 Bulgaria National Olympiad, 1
Tags: number theory , divides , divisible
Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

2011 QEDMO 9th, 6
Tags: number theory , divides , divisible
Show that there are infinitely many pairs $(m, n)$ of natural numbers $m, n \ge 2$, for $m^m- 1$ is divisible by $n$ and $n^n- 1$ is divisible by $m$.

2013 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , divides , divisible
Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns

2019 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , divides , Sum
Let $n$ be a nonnegative integer and $M =\{n^3, n^3+1, n^3+2, ..., n^3+n\}$. Consider $A$ and $B$ two nonempty, disjoint subsets of $M$ such that the sum of elements of the set $A$ divides the sum of elements of the set $B$. Prove that the number of elements of the set $A$ divides the number of elements of the set $B$.

2022 IFYM, Sozopol, 1
Tags: number theory , divides
Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

2013 Balkan MO Shortlist, A5
Tags: divides , polynomial , algebra , Integer Polynomial
Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.

2018 Thailand Mathematical Olympiad, 5
Tags: number theory , divides , minimum
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?

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