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: 15460

2016 JBMO Shortlist, 2
Tags: number theory , divide
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions: a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.

2011 Postal Coaching, 6
Tags: number theory
A positive integer is called [i]monotonic[/i] if when written in base $10$, the digits are weakly increasing. Thus $12226778$ is monotonic. Note that a positive integer cannot have first digit $0$. Prove that for every positive integer $n$, there is an $n$-digit monotonic number which is a perfect square.

2016 Postal Coaching, 5
Tags: combinatorics , number theory
For even positive integer $n$ we put all numbers $1, 2, \cdots , n^2$ into the squares of an $n \times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that it is possible to have $\frac{S_1}{S_2}=\frac{39}{64}$.

2019 European Mathematical Cup, 1
Tags: number theory , combinatorics
Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule: $$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$ Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$. [i]Proposed by Ivan Novak[/i]

2017 Dutch BxMO TST, 1
Tags: combinatorics , number theory
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

2010 Romania National Olympiad, 4
Tags: number theory
Let $a,b,c,d$ be positive integers, and let $p=a+b+c+d$. Prove that if $p$ is a prime, then $p$ is not a divisor of $ab-cd$. [i]Marian Andronache[/i]

1977 IMO Longlists, 37
Tags: number theory
Let $A_1,A_2,\ldots ,A_{n+1}$ be positive integers such that $(A_i,A_{n+1})=1$ for every $i=1,2,\ldots ,n$. Show that the equation \[x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} }\] has an infinite set of solutions $(x_1,x_2,\ldots , x_{n+1})$ in positive integers.

2018 Regional Olympiad of Mexico Center Zone, 1
Tags: number theory , digit , palindrome
Let $M$ and $N$ be two positive five-digit palindrome integers, such that $M <N$ and there is no other palindrome number between them. Determine the possible values ​​of $N-M$.

2010 Costa Rica - Final Round, 4
Tags: number theory , inequalities
Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

2009 Croatia Team Selection Test, 4
Tags: number theory
Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

1984 IMO Longlists, 55
Tags: modular arithmetic , number theory
Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers. $(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number. $(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.

PEN A Problems, 73
Tags: number theory , divisibility theory
Determine all pairs $(n,p)$ of positive integers such that [list][*] $p$ is a prime, $n>1$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

2020 Greece National Olympiad, 3
Tags: combinatorics , number theory
On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order. We have the right of ''movement'' $K$: [i]We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$.[/i] We repeat the movement $K$, many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i) $2020^{2020}$ (ii)$2021^{2020}$

2025 Harvard-MIT Mathematics Tournament, 1
Tags: algebra , number theory
Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1.$

2019 IFYM, Sozopol, 5
Tags: number theory , divisibility , greatest common divisor , modulo
Prove that there exist a natural number $a$, for which 999 divides $2^{5n}+a.5^n$ for $\forall$ odd $n\in \mathbb{N}$ and find the smallest such $a$.

2022 Junior Macedonian Mathematical Olympiad, P1
Tags: number theory , diophantine equation , factorial
Determine all positive integers $a$, $b$ and $c$ which satisfy the equation $$a^2+b^2+1=c!.$$ [i]Proposed by Nikola Velov[/i]

1967 IMO Shortlist, 1
Tags: algebra , decimal representation , perfect cube , number theory
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

2024 UMD Math Competition Part II, #2
Tags: number theory
Consider a set $S = \{a_1, \ldots, a_{2024}\}$ consisting of $2024$ distinct positive integers that satisfies the following property: [center] "For every positive integer $m < 2024,$ the sum of no $m$ distinct elements of $S$ is a multiple of $2024.$" [/center] Prove $a_1, \ldots, a_{2024}$ all leave the same remainder when divided by $2024.$ Justify your answer.

2015 Saudi Arabia JBMO TST, 1
Tags: number theory , digit , divisible
A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$

2010 Contests, 1
Tags: modular arithmetic , number theory
[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

2020 Purple Comet Problems, 28
Tags: number theory
Let $p, q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.

2013 India PRMO, 1
Tags: number theory , minimum
What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

1978 Canada National Olympiad, 1
Tags: modular arithmetic , number theory
Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?

2019 Miklós Schweitzer, 3
Tags: number theory
Prove that there are infinitely many integers $m$, $n$, such that $1 < m < n$, and the greatest common divisors $(m, n)$, $(m, n+1)$, $(m+1, n)$ and $(m+1, n+1)$ are all greater than $\sqrt{n}/999$.

2024 Serbia Team Selection Test, 4
Tags: number theory
Let $n!_0$ denote the number obtained from $n!$ by deleting all the zeroes in the end of it decimal representation. Find all positive integers $a, b, c$, such that $a!_0+b!_0=c!_0$.