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: 1362

2005 MOP Homework, 5
Tags: number theory unsolved , number theory
Does there exist an infinite subset $S$ of the natural numbers such that for every $a$, $b \in S$, the number $(ab)^2$ is divisible by $a^2-ab+b^2$?

2011 Estonia Team Selection Test, 2
Tags: number theory unsolved , number theory
Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

1999 Brazil Team Selection Test, Problem 5
Tags: quadratics , number theory unsolved , number theory
(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

1986 IMO Longlists, 62
Tags: number theory unsolved , number theory
Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$, where $p$ is a given prime number.

2006 China Northern MO, 2
Tags: induction , number theory unsolved , number theory
$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$. Show that if $a_{1}=5$, the $16 \mid a_{81}$.

2014 Spain Mathematical Olympiad, 2
Tags: number theory unsolved , number theory
Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.

1970 IMO Longlists, 2
Tags: modular arithmetic , Euler , number theory unsolved , number theory
Prove that the two last digits of $9^{9^{9}}$ and $9^{9^{9^{9}}}$ are the same in decimal representation.

2007 Tournament Of Towns, 4
Tags: number theory unsolved , number theory
A binary sequence is constructed as follows. If the sum of the digits of the positive integer $k$ is even, the $k$-th term of the sequence is $0$. Otherwise, it is $1$. Prove that this sequence is not periodic.

2007 Pre-Preparation Course Examination, 9
Tags: modular arithmetic , number theory unsolved , number theory
Solve the equation $4xy-x-y=z^2$ in positive integers.

1994 APMO, 3
Tags: number theory , relatively prime , prime numbers , number theory unsolved
Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

2005 All-Russian Olympiad, 2
Tags: pigeonhole principle , number theory unsolved , number theory
Lesha put numbers from 1 to $22^2$ into cells of $22\times 22$ board. Can Oleg always choose two cells, adjacent by the side or by vertex, the sum of numbers in which is divisible by 4?

2009 Kyrgyzstan National Olympiad, 5
Tags: induction , floor function , number theory unsolved , number theory
Prove for all natural $n$ that $\left. {{{40}^n} \cdot n!} \right|(5n)!$

2007 Pre-Preparation Course Examination, 17
Tags: number theory unsolved , number theory
For a positive integer $n$, denote $rad(n)$ as product of prime divisors of $n$. And also $rad(1)=1$. Define the sequence $\{a_i\}_{i=1}^{\infty}$ in this way: $a_1 \in \mathbb N$ and for every $n \in \mathbb N$, $a_{n+1}=a_n+rad(a_n)$. Prove that for every $N \in \mathbb N$, there exist $N$ consecutive terms of this sequence which are in an arithmetic progression.

2011 Romania Team Selection Test, 1
Tags: number theory unsolved , number theory
Show that there are infinitely many positive integer numbers $n$ such that $n^2+1$ has two positive divisors whose difference is $n$.

2007 Balkan MO, 3
Tags: algebra , polynomial , floor function , induction , number theory unsolved , number theory
Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

2005 Morocco TST, 1
Tags: quadratics , number theory , greatest common divisor , LaTeX , number theory unsolved
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.

2012 China Girls Math Olympiad, 7
Tags: search , analytic geometry , number theory , number theory unsolved
Let $\{a_n\}$ be a sequence of nondecreasing positive integers such that $\textstyle\frac{r}{a_r} = k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that $\textstyle\frac{s}{a_s} = k$.

2011 Estonia Team Selection Test, 2
Tags: number theory unsolved , number theory
Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

2007 China Team Selection Test, 2
Tags: algebra , polynomial , Euler , number theory unsolved , number theory
After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

2007 Germany Team Selection Test, 1
Tags: algebra , polynomial , number theory unsolved , number theory
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

1998 Polish MO Finals, 2
Tags: induction , number theory unsolved , number theory
$F_n$ is the Fibonacci sequence $F_0 = F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$. Find all pairs $m > k \geq 0$ such that the sequence $x_0, x_1, x_2, ...$ defined by $x_0 = \frac{F_k}{F_m}$ and $x_{n+1} = \frac{2x_n - 1}{1 - x_n}$ for $x_n \not = 1$, or $1$ if $x_n = 1$, contains the number $1$

2003 Korea - Final Round, 2
Tags: number theory unsolved , number theory
For a positive integer, $m$, answer the following questions. 1) Show that $2^{m+1}+1$ is a prime number, when $2^{m+1}+1$ is a factor of $3^{2^m}+1$. 2) Is converse of 1) true?

2001 Tournament Of Towns, 7
Tags: number theory unsolved , number theory
It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4?

2011 Korea - Final Round, 1
Tags: number theory unsolved , number theory
Prove that there is no positive integers $x,y,z$ satisfying \[ x^2 y^4 - x^4 y^2 + 4x^2 y^2 z^2 +x^2 z^4 -y^2 z^4 =0 \]

2005 Finnish National High School Mathematics Competition, 4
Tags: modular arithmetic , number theory unsolved , number theory
The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer. Show that permuting the order of digits one can obtain an integer divisible by $7.$