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

2014 ELMO Shortlist, 5
Tags: inequalities , induction , number theory , number theory unsolved
Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2004 Croatia Team Selection Test, 1
Tags: quadratics , algebra , number theory unsolved , number theory
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

2000 Hungary-Israel Binational, 1
Tags: number theory unsolved , number theory
Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we define $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$

2006 APMO, 1
Tags: number theory unsolved , number theory
Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$) with the following property: whenever $a_1,a_2,...,a_n$ are real numbers such that $a_1+a_2+\cdots +a_n$ is an integer, there exists some $i$ such that $\left|a_i-\frac{1}{2}\right|\ge f(n)$.

2005 Korea National Olympiad, 3
Tags: number theory unsolved , number theory
For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.

2010 Nordic, 4
Tags: number theory unsolved , number theory
A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n = a_1 \pm a_2 \pm a_3 \pm \cdots \pm a_k$ where $a_1, \dots , a_k$ are simple.

2019 IFYM, Sozopol, 6
Tags: function , number theory unsolved , number theory
Does there exist a function $f: \mathbb N \to \mathbb N$ such that for all integers $n \geq 2$, \[ f(f(n-1)) = f (n+1) - f(n)\, ?\]

2003 France Team Selection Test, 1
Tags: analytic geometry , modular arithmetic , number theory unsolved , number theory
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

1976 IMO Longlists, 14
Tags: calculus , integration , number theory unsolved , number theory
A sequence $\{ u_n \}$ of integers is defined by \[u_1 = 2, u_2 = u_3 = 7,\] \[u_{n+1} = u_nu_{n-1} - u_{n-2}, \text{ for }n \geq 3\] Prove that for each $n \geq 1$, $u_n$ differs by $2$ from an integral square.

2017 Bundeswettbewerb Mathematik, 4
Tags: number theory , number theory unsolved , Sequence , recursion , prime numbers , Integer
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$: (a) The number $a_n$ is a positive integer. (b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$. (c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.

1991 China Team Selection Test, 2
Tags: function , number theory unsolved , number theory
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions: (1) $f(0) = 0, f(1) = 1,$ (2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$ Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$

2005 China Team Selection Test, 3
Tags: Euler , floor function , number theory unsolved , number theory
Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\] Prove that \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

2015 Spain Mathematical Olympiad, 2
Tags: number theory unsolved , number theory
Let $p$ and $n$ be a natural numbers such that $p$ is a prime and $1+np$ is a perfect square. Prove that the number $n+1$ is sum of $p$ perfect squares.

2013 Canada National Olympiad, 2
Tags: modular arithmetic , number theory unsolved , number theory
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?

2008 Tuymaada Olympiad, 2
Tags: AMC , USA(J)MO , USAMO , number theory , prime numbers , number theory unsolved
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

1992 USAMO, 1
Tags: function , induction , geometric series , number theory unsolved , number theory
Find, as a function of $\, n, \,$ the sum of the digits of \[ 9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right), \] where each factor has twice as many digits as the previous one.

1982 IMO Longlists, 26
Tags: number theory unsolved , number theory
Let $(a_n)_{n\geq0}$ and $(b_n)_{n \geq 0}$ be two sequences of natural numbers. Determine whether there exists a pair $(p, q)$ of natural numbers that satisfy \[p < q \quad \text{ and } \quad a_p \leq a_q, b_p \leq b_q.\]

2013 India Regional Mathematical Olympiad, 4
Tags: algebra , polynomial , quadratics , modular arithmetic , number theory unsolved , number theory
A polynomial is called Fermat polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000$. Prove that $f(x)+2x$ is not a fermat polynomial

2020 Bundeswettbewerb Mathematik, 2
Tags: number theory , number theory unsolved , rational number , algebra , system of equations
Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.

2005 Canada National Olympiad, 2
Tags: inequalities , number theory unsolved , number theory
Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$. $a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$. $b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.

2011 South africa National Olympiad, 5
Tags: function , inequalities , number theory , relatively prime , number theory unsolved
Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties: [list] [*] $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$ [*] $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$[/list]

1995 Canada National Olympiad, 4
Tags: number theory , Diophantine equation , number theory unsolved
Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.

2012 Brazil National Olympiad, 4
Tags: floor function , number theory unsolved , number theory
There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that \[ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} \] where $p_i$ is the i-th prime ($p_1=2,p_2=3,p_3=5,p_4=7,\ldots$) and $a_i>1$ for all $i$?

1989 Vietnam National Olympiad, 1
Tags: number theory unsolved , number theory
Are there integers $ x$, $ y$, not both divisible by $ 5$, such that $ x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}$?

2003 India National Olympiad, 4
Tags: modular arithmetic , number theory unsolved , number theory
Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$.