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

2006 Federal Math Competition of S&M, Problem 3
Tags: number theory
For every natural number $a$, consider the set $S(a)=\{a^n+a+1|n=2,3,\ldots\}$. Does there exist an infinite set $A\subset\mathbb N$ with the property that for any two distinct elements $x,y\in A$, $x$ and $y$ are coprime and $S(x)\cap S(y)=\emptyset$?

2014 Romania Team Selection Test, 2
Tags: modular arithmetic , algebra , number theory , induction
Let $n \ge 2$ be an integer. Show that there exist $n+1$ numbers $x_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}$, so that $\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}$, where $\{ x \}$ is the fractionary part of $x$.

2008 Irish Math Olympiad, 4
Tags: number theory , linear algebra , matrix
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and $ x_1\plus{}...\plus{}x_n \equiv k$ mod 4 a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$. (b) Prove that $ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$ for all positive integers $ n$.

2022 Junior Balkan Team Selection Tests - Moldova, 7
Tags: number theory
A program works as follows. If the input is given a natural number $n$ ($n \ge 2$), then the program consecutively performs the following procedure: it determines the greatest proper divisor of the number $ n$ (that is, different from $1$ and $n$) and subtracts it from the number $n$, then applies again the same procedure to the obtained result and so on. If the program cannot find any proper divisor of the given number at a step, then it stops and outputs the total number $m$ of procedures performed (this number can be equal to $0$). The input was given the number $n = 13^{13}$. Determine the respective number $m$ at the output.

2008 CHKMO, 2
Tags: polynomial , modular arithmetic , number theory , algebra
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

1997 Federal Competition For Advanced Students, P2, 1
Tags: number theory , linear algebra , matrix
Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system: $ 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a,$ $ (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1,$ $ (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.$

2003 Greece Junior Math Olympiad, 4
Tags: number theory
Find all positive integers which can be written in the form $(mn+1)/(m+n)$, where $m,n$ are positive integers.

2015 Denmark MO - Mohr Contest, 2
Tags: divide , number theory
The numbers $1, 2, 3, . . . , 624$ are paired in such a way that the sum of the two numbers in each pair is $625$. For example $1$ and $624$ form a pair, and $30$ and $595$ form a pair. In how many of the $312$ pairs does the smaller number evenly divide the larger?

2008 ISI B.Math Entrance Exam, 10
Tags: greatest common divisor , number theory
If $p$ is a prime number and $a>1$ is a natural number , then show that the greatest common divisor of the two numbers $a-1$ and $\frac{a^p-1}{a-1}$ is either $1$ or $p$ .

2019 LIMIT Category A, Problem 7
Tags: number theory , combinatorics
How many six-digit perfect squares can be formed using all the numbers $1,2,3,4,5,6$ as digits? $\textbf{(A)}~5$ $\textbf{(B)}~19$ $\textbf{(C)}~7$ $\textbf{(D)}~\text{None of the above}$

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 $1 \le x \le 16$?

2009 USAMO, 6
Tags: number theory , greatest common divisor
Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.

2020 BMT Fall, Tie 3
Tags: number theory
Three distinct integers $a_1$, $a_2$, $a_3$ between $1$ and $21$, inclusive, are selected uniformly at random. The probability that the greatest common factor of $a_i-a_j$ and $21$ is $7$ for some positive integers $i $ and $j$, where $1 \le i \ne j \le3 $, can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2007 South East Mathematical Olympiad, 3
Tags: number theory , function
Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$, determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$, where $n\ge 2$ . ($[x]$ denotes the greatest integer not exceeding x)

2009 JBMO Shortlist, 2
Tags: number theory
A group of $n > 1$ pirates of different age owned total of $2009$ coins. Initially each pirate (except the youngest one) had one coin more than the next younger. a) Find all possible values of $n$. b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If $n = 7$, find the largest possible number of coins a pirate can have after several days.

1999 Mongolian Mathematical Olympiad, Problem 1
Tags: number theory
Prove that for any $n$ there exists a positive integer $k$ such that all the numbers $k\cdot2^s+1~(s=1,\ldots,n)$ are composite.

2016 Romania Team Selection Tests, 1
Tags: number theory
Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers

2018 Azerbaijan IMO TST, 2
Tags: number theory
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

2012 QEDMO 11th, 1
Tags: diophantine , number theory , diophantine equation
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.

1956 AMC 12/AHSME, 31
Tags: number theory , modular arithmetic
In our number system the base is ten. If the base were changed to four you would count as follows: $ 1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be: $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 38 \qquad\textbf{(C)}\ 44 \qquad\textbf{(D)}\ 104 \qquad\textbf{(E)}\ 110$

2019 Belarus Team Selection Test, 4.2
Tags: diophantine , number theory
Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2019 Moldova Team Selection Test, 1
Tags: number theory , easy
Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$. (A natural number is a positive integer)

LMT Accuracy Rounds, 2022 S10
Tags: combinatorics , number theory
In a room, there are $100$ light switches, labeled with the positive integers ${1,2, . . . ,100}$. They’re all initially turned off. On the $i$ th day for $1 \le i \le 100$, Bob flips every light switch with label number $k$ divisible by $i$ a total of $\frac{k}{i}$ times. Find the sum of the labels of the light switches that are turned on at the end of the $100$th day.

2016 Bangladesh Mathematical Olympiad, 1
Tags: number theory , modular arithmetic
(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

1970 IMO Longlists, 18
Tags: product , partition , number theory , modular arithmetic
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.