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

1999 Tournament Of Towns, 1
Tags: combinatorics , divisible , divide , sum
For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

2022 IFYM, Sozopol, 1
Tags: number theory , divide
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.)

2017 Saudi Arabia BMO TST, 1
Tags: number theory , divide , divisible , factorial
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

2022 Federal Competition For Advanced Students, P2, 1
Tags: function , number theory , divide
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

2001 Estonia National Olympiad, 2
Tags: least common multiple , divide , divisor , number theory
A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

1956 Moscow Mathematical Olympiad, 323
Tags: number theory , divisor , divide
a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

2013 Saudi Arabia Pre-TST, 2.1
Tags: number theory , remainder , divide , divisible
Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

2020 Colombia National Olympiad, 4
Tags: arithmetic sequence , divide , 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.

2003 Switzerland Team Selection Test, 4
Tags: divide , max , number theory
Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

VMEO IV 2015, 10.3
Tags: number theory , divide , divisible
Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

1999 Abels Math Contest (Norwegian MO), 2b
Tags: divide , divisible , number theory
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$

2018 Istmo Centroamericano MO, 1
Tags: number theory , recurrence relation , divide , divisible
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.

1970 Poland - Second Round, 3
Tags: number theory , divide , divisible
Prove the theorem: There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.

2003 Cuba MO, 2
Tags: number theory , divide
Prove that if $$\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}$$ where $p, q \in Z_+$ then $p$ is divisible by $2003$.

2020 Silk Road, 1
Tags: inequalities , divisible , divide , number theory
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.

2015 Gulf Math Olympiad, 1
Tags: number theory , odd , divide , divisible , prime
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: 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 $P(x) - Q(x)$.

2016 Saudi Arabia BMO TST, 3
Tags: number theory , divide , divisible
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.

2000 Tournament Of Towns, 3
Tags: least common multiple , lcm , number theory , sum , product , divisible , divide
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

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

1984 Bundeswettbewerb Mathematik, 1
Tags: number theory , divisible , divide
The natural numbers $n$ and $z$ are relatively prime and greater than $1$. For $k = 0, 1, 2,..., n - 1$ let $s(k) = 1 + z + z^2 + ...+ z^k.$ Prove that: a) At least one of the numbers $s(k)$ is divisible by $n$. b) If $n$ and $z - 1$ are also coprime, then already one of the numbers $s(k)$ with $k = 0,1, 2,..., n- 2$ is divisible by $n$.

2014 India PRMO, 13
Tags: number theory , integer , divide
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?

2000 Czech And Slovak Olympiad IIIA, 1
Tags: number theory , even , divide , divisible
Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

2000 Portugal MO, 3
Tags: number theory , divide , divisor
Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2013 Saudi Arabia Pre-TST, 1.2
Tags: number theory , divide , divisible
Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.