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

2013 Hitotsubashi University Entrance Examination, 1
Tags: number theory
Find all pairs $(p,\ q)$ of positive integers such that $3p^3-p^2q-pq^2+3q^3=2013.$

1974 IMO Longlists, 35
Tags: prime number , number theory
If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$

2016 All-Russian Olympiad, 6
Tags: graph theory , number theory , combinatorics
There are $n>1$ cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges). Major of every city X counted amount of such numberings of all cities from $1$ to $n$ , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of $2016$. Prove, that result of last major is multiple of $2016$ too.

2017 India IMO Training Camp, 3
Tags: number theory
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

1999 India National Olympiad, 6
Tags: number theory
For which positive integer values of $n$ can the set $\{ 1, 2, 3, \ldots, 4n \}$ be split into $n$ disjoint $4$-element subsets $\{ a,b,c,d \}$ such that in each of these sets $a = \dfrac{b +c +d} {3}$.

1994 Bundeswettbewerb Mathematik, 2
Tags: sequence , number theory , divisibility
Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by $$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$ Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.

2010 Ukraine Team Selection Test, 12
Tags: algebra , rational , sum , reciprocal sum , number theory
Is there a positive integer $n$ for which the following holds: for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?

2019 Purple Comet Problems, 10
Tags: combinatorics , number theory
Find the number of positive integers less than $2019$ that are neither multiples of $3$ nor have any digits that are multiples of $3$.

2020 Thailand TST, 5
Tags: functional equation , number theory
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2011 Romanian Master of Mathematics, 2
Tags: algebra , polynomial , modular arithmetic , function , number theory , greatest common divisor , system of equations
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

1994 Baltic Way, 7
Tags: number theory
Let $p>2$ be a prime number and \[1+\frac{1}{2^3}+\frac{1}{3^3}+\ldots +\frac{1}{(p-1)^3}=\frac{m}{n}\] where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.

2009 Hanoi Open Mathematics Competitions, 3
Tags: number theory , perfect square
Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$ Prove that $a + b$ is a square.

2015 Serbia National Math Olympiad, 4
Tags: number theory
For integer $a$, $a \neq 0$, $v_2(a)$ is greatest nonnegative integer $k$ such that $2^k | a$. For given $n \in \mathbb{N}$ determine highest possible cardinality of subset $A$ of set $ \{1,2,3,...,2^n \} $ with following property: For all $x, y \in A$, $x \neq y$, number $v_2(x-y)$ is even.

1970 Polish MO Finals, 3
Tags: binomial coefficient , number theory , divisible , prime
Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.

2012 Moldova Team Selection Test, 5
Tags: number theory
Find all pairs $(m, n)$ of integers for which $$\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}.$$

1998 Kurschak Competition, 1
Tags: relatively prime , number theory
Is there an infinite sequence of positive integers where no two terms are relatively prime, no term divides any other term, and there is no integer larger than $1$ that divides every term of the sequence?

2011 Postal Coaching, 2
Tags: inequalities , number theory
Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that \[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\] for all $n \ge 1$.

2018 Balkan MO Shortlist, N1
Tags: number theory
For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that: [list] [*] $a_1 \geq 2018^{2018};$ [*] $a_m \leq a_n$ whenever $m \leq n$; [*] $d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$? [/list] [i](Dominic Yeo, United Kingdom)[/i]

2021 HMNT, 3
Tags: number theory
Suppose $m$ and $n$ are positive integers for which $\bullet$ the sum of the first $m$ multiples of $n$ is $120$, and $\bullet$ the sum of the first $m^3$ multiples of$ n^3$ is $4032000$. Determine the sum of the first $m^2$ multiples of $n^2$

2021 CMIMC, 2.5 1.2
Tags: algebra , number theory
Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2\text{ mod }n)$th hole, where $k\text{ mod }n$ is the remainder when $k$ is divided by $n$. What is minimum $n$ such that there is at most one pigeon per hole? [i]Proposed by Christina Yao[/i]

2020 Taiwan TST Round 2, 1
Tags: number theory , functional equation
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2012 Dutch IMO TST, 1
Tags: perfect power , number theory , prime , gcd , greatest common divisor
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2023 Turkey Olympic Revenge, 4
Tags: functional equation , number theory , revenge
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$ is a perfect square. [i]Proposed by Barış Koyuncu[/i]

2018 Saint Petersburg Mathematical Olympiad, 6
Tags: irrational number , number theory
$\alpha,\beta$ are positive irrational numbers and $[\alpha[\beta x]]=[\beta[\alpha x]]$ for every positive $x$. Prove that $\alpha=\beta$

2010 India National Olympiad, 2
Tags: floor function , number theory
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.