Olympiad problems

2015 Postal Coaching, Problem 4

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

2009 Purple Comet Problems, 13

Tags: prime number , number theory

How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?

2021 Austrian MO Regional Competition, 4

Tags: number theory , divide , divisible

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

2010 Romania Team Selection Test, 2

Tags: ceiling function , algebra , polynomial , number theory

(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square. (b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$. [i]AMM Magazine[/i]

KoMaL A Problems 2019/2020, A. 769

Tags: number theory

Find all triples $(a,b,c)$ of distinct positive integers so that there exists a subset $S$ of the positive integers for which for all positive integers $n$ exactly one element of the triple $(an,bn,cn)$ is in $S$. Proposed by Carl Schildkraut, MIT

2008 Bundeswettbewerb Mathematik, 2

Tags: inequalities , greatest common divisor , least common multiple , relatively prime , number theory

Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.

2013 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

2021 All-Russian Olympiad, 7

Tags: number theory

Find all permutations $(a_1, a_2,...,a_{2021})$ of $(1,2,...,2021)$, such that for every two positive integers $m$ and $n$ with difference bigger than $20^{21}$, the following inequality holds: $GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$.

2018 Mediterranean Mathematics OIympiad, 3

Tags: number theory , prime number

An integer $a\ge1$ is called [i]Aegean[/i], if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$. (Proposed by Gerhard Woeginger, Austria)

2018 Saudi Arabia BMO TST, 3

Tags: divide , divisor , number theory

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

2012 ELMO Shortlist, 3

Tags: number theory , additive number theory , extremal combinatorics , perfect power

Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$. [i]Alex Zhu.[/i]

2018 India PRMO, 19

Tags: number theory

Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$

2011 Romanian Master of Mathematics, 4

Tags: function , number theory

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

2003 Moldova Team Selection Test, 1

Tags: quadratic , number theory , relatively prime , diophantine equation

Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.

2006 Princeton University Math Competition, 8

Tags: number theory

Find all integers $n$ (not necessarily positive) such that $7n^3-3n^2-3n-1$ is a perfect cube.

2020 BMT Fall, 11

Tags: number theory

Compute $\sum^{999}_{x=1}\gcd (x, 10x + 9)$.

2022 Saudi Arabia JBMO TST, 1

Tags: number theory , diophantine equation

Find all pairs of positive prime numbers $(p, q)$ such that $$p^5 + p^3 + 2 = q^2 - q.$$

2019 Gulf Math Olympiad, 2

Tags: number theory , digit , multiple

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?

2020-2021 Winter SDPC, #5

Tags: number theory

Suppose that the positive divisors of a positive integer $n$ are $1=d_1<d_2<\ldots<d_k=n$, where $k \geq 5$. Given that $k \leq 1000$ and $n={d_2}^{d_3}{d_4}^{d_5}$, compute, with proof, all possible values of $k$.

2023 Poland - Second Round, 6

Tags: number theory , prime number , combinatorics , chessboard

Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.

2003 Flanders Math Olympiad, 3

Tags: number theory

A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.

2020 BMT Fall, 24

Tags: number theory

Let $N$ be the number of non-empty subsets $T$ of $S = \{1, 2, 3, 4, . . . , 2020\}$ satisfying $\max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

2022 Baltic Way, 16

Tags: number theory

Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition $$ f(a) + f(b) \mid (a + b)^2$$ for all $a,b \in \mathbb{Z^+}$

2015 Cuba MO, 4

Tags: number theory , digit

Let $A = \overline{abcd}$ be a $4$-digit positive integer, such that $a\ge 7$ and $a > b >c > d > 0$. Let us consider a positive integer $B = \overline{dcba}$. If all digits of $A+B$ are odd, determine all possible values of $A$.

2009 Federal Competition For Advanced Students, P2, 5

Tags: number theory

Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$. How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime?