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

2021 Bangladeshi National Mathematical Olympiad, 3
Tags: number theory , integer part , floor function
Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$?

2024 Belarusian National Olympiad, 10.7
Tags: number theory
Let's call a pair of positive integers $(k,n)$ [i]interesting[/i] if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$ Find the number of interesting pairs $(k,n)$ with $k \leq 100$ [i]M. Karpuk[/i]

2023 India Regional Mathematical Olympiad, 3
Tags: number theory
For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$ (s(n))^2=m \text { and }(s(m))^2=n . $$

1910 Eotvos Mathematical Competition, 2
Tags: number theory , multiple , divide
Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2019 Junior Balkan Team Selection Tests - Moldova, 5
Tags: number theory
Find all triplets of positive integers $(a, b, c)$ that verify $\left(\frac{1}{a}+1\right)\left(\frac{1}{b}+1\right)\left(\frac{1}{c}+1\right)=2$.

2022 Korea Winter Program Practice Test, 1
Tags: number theory , diophantine equation
Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

2014 Dutch Mathematical Olympiad, 4
Tags: number theory , prime , divisible , inequality , diophantine
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if - $p$ is an odd prime number, - $a, b$, and $c$ are distinct and - $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$. a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $

2015 Azerbaijan National Olympiad, 4
Tags: prime divisor , number theory , divisor
Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

2002 Federal Math Competition of S&M, Problem 3
Tags: number theory , diophantine equation
Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

2024 Bundeswettbewerb Mathematik, 2
Tags: digit , perfect square , number theory
Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

2001 Slovenia National Olympiad, Problem 2
Tags: number theory
Find all prime numbers $p$ for which $3^p-(p+2)^2$ is also prime.

2018 Belarusian National Olympiad, 9.6
Tags: inequalities , number theory
For all positive integers $m$ and $n$ prove the inequality $$ |n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1. $$

2015 India IMO Training Camp, 2
Tags: number theory
For a composite number $n$, let $d_n$ denote its largest proper divisor. Show that there are infinitely many $n$ for which $d_n +d_{n+1}$ is a perfect square.

2018 Brazil Undergrad MO, 3
Tags: number theory , permutation
How many permutations $a_1, a_2, a_3, a_4$ of $1, 2, 3, 4$ satisfy the condition that for $k = 1, 2, 3,$ the list $a_1,. . . , a_k$ contains a number greater than $k$?

2004 Tournament Of Towns, 5
Tags: number theory , digit
Two $10$-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of $10$-digit integers with no two integers being neighbours.

2009 May Olympiad, 1
Tags: digit , number theory
Each two-digit natural number is [i]assigned [/i] a digit as follows: Its digits are multiplied. If the result is a digit, this is the assigned digit. If the result is a two-digit number, these two figures are multiplied, and if the result is a digit, this is the assigned digit. Otherwise, the operation is repeated. For example, the digit assigned to $32$ is $6$ since $3 \times = 6$; the digit assigned to $93$ is $4$ since $9 \times 3 = 27$, $2 \times 7 = 14$, $1 \times 4 = 4$. Find all the two-digit numbers that are assigned $8$.

1991 Tournament Of Towns, (287) 3
Tags: number theory , digit
We are looking for numbers ending with the digit $5$ such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property. (a) Find four such numbers. (b) Prove that there are infinitely many. (A. Andjans, Riga)

2011 Irish Math Olympiad, 3
Tags: induction , modular arithmetic , number theory , greatest common divisor , algebra
The integers $a_0, a_1, a_2, a_3,\ldots$ are defined as follows: $a_0 = 1$, $a_1 = 3$, and $a_{n+1} = a_n + a_{n-1}$ for all $n \ge 1$. Find all integers $n \ge 1$ for which $na_{n+1} + a_n$ and $na_n + a_{n-1}$ share a common factor greater than $1$.

2002 All-Russian Olympiad Regional Round, 11.1
Tags: number theory , algebra , rational , irrational
The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

2024 Azerbaijan IZhO TST, 4
Tags: number theory
Take a sequence $(a_n)_{n=1}^\infty$ such that $a_1=3$ $a_n=a_1a_2a_3...a_{n-1}-1$ [b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence. [b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.

2011 ELMO Problems, 5
Tags: modular arithmetic , quadratic , number theory
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and \[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\] [i]Alex Zhu.[/i] [hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]

2023 South East Mathematical Olympiad, 4
Tags: inequalities , number theory
Find the largest real number $c$, such that for any integer $s>1$, and positive integers $m, n$ coprime to $s$, we have$$ \sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs$$ where $\{ x \} = x - \lfloor x \rfloor $.

2021 Olympic Revenge, 5
Tags: number theory , diophantine equation , arithmetic sequence , perfect square
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.

1996 Austrian-Polish Competition, 7
Tags: number theory
Prove there are no such integers $ k, m $ which satisfy $ k \ge 0, m \ge 0 $ and $ k!+48=48(k+1)^m $.

2020 Cono Sur Olympiad, 2
Tags: number theory , combinatorics , pigeonhole principle
Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.