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: 1766

2009 All-Russian Olympiad, 5
Tags: pigeonhole principle , number theory proposed , number theory
Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i\plus{}1}\minus{}a_i$.

2001 Turkey MO (2nd round), 1
Tags: number theory proposed , number theory
Find all ordered triples of positive integers $(x,y,z)$ such that \[3^{x}+11^{y}=z^{2}\]

1992 IberoAmerican, 1
Tags: number theory proposed , 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 $(a_{1992},\, b_{1992})$.

1977 IMO Longlists, 12
Tags: modular arithmetic , number theory , relatively prime , number theory proposed
Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$

2003 ITAMO, 4
Tags: number theory proposed , number theory
There are two sorts of people on an island: [i]knights[/i], who always talk truth, and [i]scoundrels[/i], who always lie. One day, the people establish a council consisting of $2003$ members. They sit around a round table, and during the council each member said: "Both my neighbors are scoundrels". In a later day, the council meets again, but one member could not come due to illness, so only $2002$ members were present. They sit around the round table, and everybody said: "Both my neighbors belong to the sort different from mine". Is the absent member a knight or a scoundrel?

2012 Romanian Masters In Mathematics, 4
Tags: modular arithmetic , function , number theory , number theory proposed
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not. [i](Russia) Valery Senderov[/i]

2007 India IMO Training Camp, 2
Tags: modular arithmetic , Diophantine equation , number theory proposed , number theory
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2007 Junior Balkan Team Selection Tests - Romania, 4
Tags: number theory proposed , number theory
We call a real number $x$ with $0 < x < 1$ [i]interesting[/i] if $x$ is irrational and if in its decimal writing the first four decimals are equal. Determine the least positive integer $n$ with the property that every real number $t$ with $0 < t < 1$ can be written as the sum of $n$ pairwise distinct interesting numbers.

2007 Baltic Way, 17
Tags: inequalities , number theory , greatest common divisor , number theory proposed
Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.

1992 Iran MO (2nd round), 1
Tags: modular arithmetic , number theory proposed , number theory
Prove that for any positive integer $t,$ \[1+2^t+3^t+\cdots+9^t - 3(1 + 6^t +8^t )\] is divisible by $18.$

2014 Taiwan TST Round 3, 2
Tags: modular arithmetic , number theory , relatively prime , number theory proposed , primitive root
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.

2003 Tournament Of Towns, 3
Tags: number theory proposed , number theory
For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

2023 China Western Mathematical Olympiad, 7
Tags: number theory , algebra , China , Inequality , number theory proposed
For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

2011 Benelux, 1
Tags: number theory proposed , number theory
An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a [i]Benelux couple[/i] if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$. (a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$. (b) Prove that there are infinitely many Benelux couples

2006 Iran Team Selection Test, 1
Tags: number theory , least common multiple , function , number theory proposed
Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

2005 ITAMO, 1
Tags: number theory proposed , number theory
Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$, but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$, find the greatest possible $n$.

2015 IFYM, Sozopol, 2
Tags: function , induction , number theory proposed , number theory
Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

2008 All-Russian Olympiad, 1
Tags: number theory proposed , number theory
Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?

2012 ELMO Shortlist, 6
Tags: floor function , algebra , polynomial , Vieta , quadratics , inequalities , number theory proposed
Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2002 Tournament Of Towns, 5
Tags: floor function , number theory proposed , number theory
An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.

2009 Pan African, 3
Tags: number theory proposed , number theory
Let $x$ be a real number with the following property: for each positive integer $q$, there exists an integer $p$, such that \[\left|x-\frac{p}{q} \right|<\frac{1}{3q}. \] Prove that $x$ is an integer.

2007 Danube Mathematical Competition, 4
Tags: number theory , prime numbers , number theory proposed
Let $ a,n$ be positive integers such that $ a\ge(n\minus{}1)!$. Prove that there exist $ n$ [i]distinct[/i] prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a\plus{}i$, for all $ i\equal{}\overline{1,\ldots,n}$.

2014 Turkey Team Selection Test, 1
Tags: modular arithmetic , number theory proposed , number theory
Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.

1998 Tournament Of Towns, 2
Tags: number theory proposed , number theory
The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundreds-digit is even.

2005 Baltic Way, 18
Tags: modular arithmetic , number theory proposed , number theory
Let $x$ and $y$ be positive integers and assume that $z=\frac{4xy}{x+y}$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4n-1$ where $n$ is a positive integer.