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: 408

VMEO I 2004, 2
Tags: fibonacci sequence , fibonacci , number theory , perfect power , fibonacci number , divisible
The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows: $$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$ Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.

2010 Cuba MO, 5
Tags: number theory , divide , divisible
Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.

2003 Austria Beginners' Competition, 3
Tags: number theory , consecutive , divide , divisible
a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

2010 Hanoi Open Mathematics Competitions, 2
Tags: number theory , divisible
Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above.

2020 Czech and Slovak Olympiad III A, 6
Tags: inequalities , binomial , number theory , divisible
For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs. (Note: A positive integer cannot begin with a digit of $0$.) (Jaromir Simsa)

1974 Dutch Mathematical Olympiad, 2
Tags: number theory , divide , divisible
$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

2019 Serbia JBMO TST, 1
Tags: number theory , divisible , factorial
Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?

2013 Saudi Arabia BMO TST, 4
Tags: divide , divisible , number theory
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.

2004 Denmark MO - Mohr Contest, 2
Tags: number theory , divide , divisible
Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

2007 Postal Coaching, 6
Tags: digit , number theory , divide , divisible
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

1994 Tournament Of Towns, (417) 5
Tags: number theory , digit , divisible , divide
Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

2001 Taiwan National Olympiad, 2
Tags: number theory , divisible
Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

2003 Singapore Senior Math Olympiad, 1
Tags: perfect square , divisible , divide , number theory
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

2010 All-Russian Olympiad Regional Round, 10.4
Tags: number theory , divisible
We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.

1979 Polish MO Finals, 1
Tags: number theory , divisible
Let be given a set $\{r_1,r_2,...,r_k\}$ of natural numbers that give distinct remainders when divided by a natural number $m$. Prove that if $k > m/2$, then for every integer $n$ there exist indices $i$ and $j$ (not necessarily distinct) such that $r_i +r_j -n$ is divisible by $m$.

2014 Czech-Polish-Slovak Junior Match, 4
Tags: number theory , divide , divisible , digit
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

2021 Dutch IMO TST, 3
Tags: number theory , divide , divisible
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

2013 Saudi Arabia Pre-TST, 4.2
Tags: number theory , divide , divisible
Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

2019 Saudi Arabia IMO TST, 2
Tags: algebra , polynomial , divisible
Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$

2000 Chile National Olympiad, 3
Tags: number theory , divide , divisible , periodic
A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$

2018 Hanoi Open Mathematics Competitions, 11
Tags: number theory , divisible
Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.

2015 Thailand Mathematical Olympiad, 1
Tags: divide , recurrence relation , divisible , number theory
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

2000 Kazakhstan National Olympiad, 5
Tags: prime divisor , divide , divisible , number theory
Let the number $ p $ be a prime divisor of the number $ 2 ^ {2 ^ k} + 1 $. Prove that $ p-1 $ is divisible by $ 2 ^ {k + 1} $.

1993 Swedish Mathematical Competition, 1
Tags: sum of digits , divisible , divide , number theory
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.

2000 Tournament Of Towns, 2
Tags: number theory , divide , divisible
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)