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

2021 Durer Math Competition Finals, 14
Tags: number theory , divisible , divide
How many functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\}$ have the property that $f(f(x))-4x$ is divisible by $17$ for all integers $1 \le x \le 16$?

2003 Austrian-Polish Competition, 7
Tags: number theory , divide , factorial , divisible
Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $. Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

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

2022 Austrian MO National Competition, 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]

2022 IFYM, Sozopol, 5
Tags: algebra , divide
Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.

2018 Bosnia and Herzegovina EGMO TST, 2
Tags: positive integer , number theory , divide
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

2001 Estonia National Olympiad, 2
Tags: max , number theory , divide
Find the maximum value of $k$ for which one can choose $k$ integers out of $1,2... ,2n$ so that none of them divides another one.

2008 Postal Coaching, 3
Tags: number theory , divide , 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.

2018 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible , functional
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

2020 Durer Math Competition Finals, 6
Tags: number theory , divide
Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

2015 Dutch BxMO/EGMO TST, 1
Tags: divisor , divide , number theory
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2012 China Northern MO, 8
Tags: number theory , divide
Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that : (1) $(a+1, \frac{a^p+1}{a+1}) = p$ (2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$. (3) $p!|(a +1) $.

1972 Spain Mathematical Olympiad, 7
Tags: number theory , multiple , divide , divisible
Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

1955 Moscow Mathematical Olympiad, 292
Tags: number theory , divide
Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$. Prove that if $n > 3$, then $6$ divides $ab$.

2013 QEDMO 13th or 12th, 2
Tags: binomial , number theory , divisible , divide
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 $p^q$.

2011 Argentina National Olympiad, 5
Tags: number theory , divide , divisible
Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.

1998 Estonia National Olympiad, 4
Tags: divide , number theory , divisible , prime
Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

2013 Saudi Arabia Pre-TST, 3.2
Tags: number theory , divide , divisible
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.

1996 Singapore Team Selection Test, 3
Tags: sequence , number theory , divide , divisible
Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$

2003 May Olympiad, 3
Tags: number theory , divide , multiple
Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.

2011 QEDMO 10th, 3
Tags: multiple , number theory , divide
Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.

2010 Saudi Arabia Pre-TST, 1.3
Tags: divide , digit , divisible , number theory
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.

2014 Saudi Arabia GMO TST, 2
Tags: sum of powers , number theory , divide , divisible
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$

2010 Thailand Mathematical Olympiad, 6
Tags: number theory , prime , divide
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

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