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

2012 Iran MO (3rd Round), 2
Tags: floor function , number theory
Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition: \[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\] [i]Proposed by Mohammad Gharakhani[/i]

2015 Belarus Team Selection Test, 3
Tags: floor function , sequence , parity , number theory
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2014 Contests, 2
Tags: fraction , number theory
Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

2016 Latvia Baltic Way TST, 17
Tags: prime , diophantine equation , diophantine , number theory
Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

2021 Lusophon Mathematical Olympiad, 1
Tags: number theory
Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$. a) Show that $super sextalternados$ numbers don't exist. b) Find the smallest $sextalternado$ number.

1994 IMO, 3
Tags: function , number theory , equation , binary representation
For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

2006 Pan African, 4
Tags: number theory , floor function
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.

2024 CAPS Match, 6
Tags: number theory , combinatorics , positive integer , combinatorial number theory , prime number , triplets
Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

VI Soros Olympiad 1999 - 2000 (Russia), 11.7
Tags: number theory , coprime
Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.

1999 Bundeswettbewerb Mathematik, 2
Tags: number theory , number theory with sequences , sequence , coprime
The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime

2024 Bulgarian Winter Tournament, 12.4
Tags: number theory
Call a positive integer $m$ $\textit{good}$ if there exist integers $a, b, c$ satisfying $m=a^3+2b^3+4c^3-6abc$. Show that there exists a positive integer $n<2024$, such that for infinitely many primes $p$, the number $np$ is $\textit{good}$.

1999 Singapore Senior Math Olympiad, 1
Tags: diophantine , number theory , diophantine equation
Find all the integral solutions of the equation $\left( 1+\frac{1}{x}\right)^{x+1}=\left( 1+\frac{1}{1999}\right)^{1999}$

1981 Brazil National Olympiad, 5
Tags: combinatorics , greatest common divisor , game strategy , number theory
Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.

2017-IMOC, N4
Tags: p-adic , number theory
Find all integers $n$ such that $n^{n-1}-1$ is square-free.

1966 IMO Longlists, 34
Tags: number theory , equation
Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

2022 District Olympiad, P3
Tags: number theory , induction , diophantine equation
$a)$ Solve over the positive integers $3^x=x+2.$ $b)$ Find pairs $(x,y)\in\mathbb{N}\times\mathbb{N}$ such that $(x+3^y)$ and $(y+3^x)$ are consecutive.

2012 Lusophon Mathematical Olympiad, 5
Tags: number theory , combinatorics
5)Players $A$ and $B$ play the following game: a player writes, in a board, a positive integer $n$, after this they delete a number in the board and write a new number where can be: i)The last number $p$, where the new number will be $p - 2^k$ where $k$ is greatest number such that $p\ge 2^k$ ii) The last number $p$, where the new number will be $\frac{p}{2}$ if $p$ is even. The players play alternately, a player win(s) if the new number is equal to $0$ and player $A$ starts. a)Which player has the winning strategy with $n = 40$?? b)Which player has the winning strategy with $n = 2012$??

2023 Germany Team Selection Test, 3
Tags: number theory
Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

1976 IMO Longlists, 38
Tags: number theory , floor function
Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.

2017 Regional Olympiad of Mexico Southeast, 4
Tags: factorial , diophantine equation , number theory
Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

TNO 2023 Senior, 2
Tags: number theory
Find all integers \( n > 1 \) such that all prime divisors of \( n^6 - 1 \) divide \( (n^2 - 1)(n^3 - 1) \).

2014 China Team Selection Test, 4
Tags: number theory , vieta jumping
Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.

2010 Balkan MO Shortlist, N3
Tags: function , greatest common divisor , relatively prime , totient function , number theory
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2005 Thailand Mathematical Olympiad, 10
Tags: remainder , sum , number theory
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?

2020/2021 Tournament of Towns, P1
Tags: number theory
Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]