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.

AND:
OR:
NO:

Found problems: 15460

2010 Baltic Way, 17
Tags: number theory , modular arithmetic
Find all positive integers $n$ such that the decimal representation of $n^2$ consists of odd digits only.

2016 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory
Let $n$ be a positive integer and consider the system \begin{align*} S(n):\begin{cases} x^2+ny^2=z^2\\ nx^2+y^2=t^2 \end{cases}, \end{align*} where $x,y,z$, and $t$ are naturals. If [list] [*] $M_1=\{n\in\mathbb N:$ system $S(n)$ has infinitely many solutions$\}$, and [*] $M_1=\{n\in\mathbb N:$ system $S(n)$ has no solutions$\}$, [/list] prove that [list] [*] $7 \in M_1$ and $10 \in M_2$. [*] sets $M_1$ and $M_2$ are infinite. [/list]

2001 Cono Sur Olympiad, 2
Tags: number theory
Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$.

2010 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: number theory , divisibility , odd
Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

2016 Kosovo National Mathematical Olympiad, 3
Tags: number theory
Show that the sum $S=5+5^2+5^3+…+5^{2016}$ is divisible by $31$

2017 Princeton University Math Competition, A6/B8
Tags: number theory
Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.

2010 Indonesia MO, 4
Tags: number theory
Given that $m$ and $n$ are positive integers with property: \[(mn)\mid(m^{2010}+n^{2010}+n)\] Show that there exists a positive integer $k$ such that $n=k^{2010}$ [i]Nanang Susyanto, Yogyakarta[/i]

2007 China Team Selection Test, 3
Tags: number theory , least common multiple , relatively prime
Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

DMM Individual Rounds, 2006 Tie
Tags: number theory , algebra
[b]p1.[/b] Suppose that $a$, $b$, and $c$ are positive integers such that not all of them are even, $a < b$, $a^2 + b^2 = c^2$, and $c - b = 289$. What is the smallest possible value for $c$? [b]p2.[/b] If $a, b > 1$ and $a^2$ is $11$ in base $b$, what is the third digit from the right of $b^2$ in base $a$? [b]p3.[/b] Find real numbers $a, b$ such that $x^2 - x - 1$ is a factor of $ax^{10} + bx^9 + 1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2005 Grigore Moisil Urziceni, 2
Tags: number theory , geometric sequence , inequalities
Find all triples $ (x,y,z) $ of natural numbers that are in geometric progression and verify the inequalities $$ 4016016\le x<y<z\le 4020025. $$

2016 USA TSTST, 4
Tags: function , number theory , euler , totient function
Suppose that $n$ and $k$ are positive integers such that \[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \] Prove that $n \le 3^k$. Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$. [i]Proposed by Linus Hamilton[/i]

2014 Finnish National High School Mathematics, 4
Tags: point , circles , prime , number theory
The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

1983 Brazil National Olympiad, 5
Tags: positive integer , minimum , inequality , number theory
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.

2015 IMO Shortlist, N6
Tags: function , periodic function , arrows , number theory
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

1994 Greece National Olympiad, 4
Tags: multiple , number theory
How many sums $$x_1+x_2+x_3, \ \ 1\leq x_j\leq 300, \ j=1,2,3$$ are multiples of $3$;

2021 Korea Winter Program Practice Test, 4
Tags: number theory , integer polynomial
Find all $f(x)\in \mathbb Z (x)$ that satisfies the following condition, with the lowest degree. [b]Condition[/b]: There exists $g(x),h(x)\in \mathbb Z (x)$ such that $$f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)$$.

2020 Taiwan APMO Preliminary, P4
Tags: number theory , algebra
Let $(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}$ all be positive interger solutions that satisfies $$1\leq a\leq b$$ and $$\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}$$ And the value of $a_n$ is [b]only[/b] determined by the following recurrence relation:$ a_{n+2} = pa_{n+1} + qa_n + r$ Find $(p,q,r)$.

2006 AMC 8, 16
Tags: least common multiple , ratio , number theory
Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read? $ \textbf{(A)}\ 6400 \qquad \textbf{(B)}\ 6600 \qquad \textbf{(C)}\ 6800 \qquad \textbf{(D)}\ 7000 \qquad \textbf{(E)}\ 7200$

2020 Turkey MO (2nd round), 4
Tags: prime number , number theory
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.

2002 Iran MO (3rd Round), 3
Tags: number theory
$a_{n}$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3$, and \[a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}\] Prove that for each natural $n$, $a_{n}$ is integer.

1992 IberoAmerican, 1
Tags: number theory
Let $\{a_{n}\}_{n \geq 0}$ and $\{b_{n}\}_{n \geq 0}$ be two sequences of integer numbers such that: i. $a_{0}=0$, $b_{0}=8$. ii. For every $n \geq 0$, $a_{n+2}=2a_{n+1}-a_{n}+2$, $b_{n+2}=2b_{n+1}-b_{n}$. iii. $a_{n}^{2}+b_{n}^{2}$ is a perfect square for every $n \geq 0$. Find at least two values of the pair $(a_{1992},\, b_{1992})$.

2006 AIME Problems, 3
Tags: number theory , integration
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer.

2016 Iran Team Selection Test, 1
Tags: number theory , divisibility
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

1950 Polish MO Finals, 6
Tags: construction , number theory
Prove that if a natural number $n$ is greater than $4$ and is not a prime number, then the produxt of the consecutive natural numbers from $1$ to $n-1$ is divisible by $ n$.

2021 Princeton University Math Competition, A1
Tags: number theory
Prove that, for an arbitrary positive integer $n \in Z_{>0}$, the number $n^2- n + 1$ does not have any prime factors of the form $6k + 5$, for $k \in Z_{>0}$.