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

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 $11 | a_t$.

2019 New Zealand MO, 1
Tags: number theory , divisible
How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both?

OMMC POTM, 2022 1
Tags: number theory , divisible
The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered? [i]Proposed by Evan Chang (squareman), USA[/i]

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

2005 Thailand Mathematical Olympiad, 19
Tags: algebra , polynomial , divisible
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.

1983 Austrian-Polish Competition, 8
Tags: number theory , recurrence relation , sequence , divide , factorial , divisible , product
(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

2014 Switzerland - Final Round, 5
Tags: number theory , divide , divisible
Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have: $$\sum_{d | n} a_d = 2^n.$$ Show for every $n \in N$ that $n$ divides $a_n$. Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$

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 $N_1, N_2, N_3,...$

2010 Thailand Mathematical Olympiad, 3
Tags: number theory , divisible , divide
Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

1972 Putnam, A5
Tags: number theory , divisible
Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

1982 Tournament Of Towns, (015) 1
Tags: number theory , divisible , divisor
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors. (M Levin)

1992 Tournament Of Towns, (341) 3
Tags: sum of digits , divisible , number theory
Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd. (D Fomin, St Petersburg)

2000 Bundeswettbewerb Mathematik, 2
Tags: number theory , sum , divisible
A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?

2019 Romanian Master of Mathematics Shortlist, C3
Tags: combinatorics , divisible , permutation
Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

1955 Moscow Mathematical Olympiad, 290
Tags: number theory , divisible , divide
Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?

2021 Polish Junior MO Second Round, 3
Tags: number theory , divisible , divide
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.

2019 Auckland Mathematical Olympiad, 2
Tags: number theory , divisible , divide
Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.

2015 Indonesia MO Shortlist, N6
Tags: number theory , divisible , set , subset
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.

2000 All-Russian Olympiad Regional Round, 9.2
Tags: number theory , divisible , divide
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 $a$?

2019 Saudi Arabia Pre-TST + Training Tests, 5.1
Tags: number theory , prime , divide , divisible
Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

2003 Chile National Olympiad, 5
Tags: number theory , divide , divisible
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)

2023 Francophone Mathematical Olympiad, 4
Tags: number theory , divide , divisible
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.

1997 Singapore MO Open, 2
Tags: binomial , divide , divisible , number theory
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

2012 Switzerland - Final Round, 7
Tags: number theory , divide , divisible
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.

1960 Poland - Second Round, 4
Tags: number theory , divide , divisible
Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.