Olympiad problems

2019 China Team Selection Test, 6

Tags: number theory

Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.

2023 Swedish Mathematical Competition, 6

Tags: algebra , number theory

Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.

2020 Baltic Way, 19

Tags: divisor function , number theory

Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.

2022 Ecuador NMO (OMEC), 4

Tags: average , number theory , combinatorics

Find the number of sets with $10$ distinct positive integers such that the average of its elements is less than 6.

2025 Thailand Mathematical Olympiad, 1

Tags: number theory

For each positive integer $m$, denote by $d(m)$ the number of positive divisors of $m$. We say that a positive integer $n$ is [i]Burapha[/i] integer if it satisfy the following condition [list] [*] $d(n)$ is an odd integer. [*] $d(k) \leqslant d(\ell)$ holds for every positive divisor $k, \ell$ of $n$, such that $k < \ell$ [/list] Find all Burapha integer.

2000 Singapore Senior Math Olympiad, 3

Tags: number theory , inequalities , sum , lcm

Let $n_1,n_2,n_3,...,n_{2000}$ be $2000$ positive integers satisfying $n_1<n_2<n_3<...<n_{2000}$. Prove that $$\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}$$ where $[a, b]$ denotes the least common multiple of $a$ and $b$.

2014 Belarus Team Selection Test, 4

Tags: number theory , diophantine , diophantine equation

Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

2006 AMC 8, 16

Tags: ratio , number theory , least common multiple

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read? $ \textbf{(A)}\ 6400 \qquad \textbf{(B)}\ 6600 \qquad \textbf{(C)}\ 6800 \qquad \textbf{(D)}\ 7000 \qquad \textbf{(E)}\ 7200$

2007 Junior Tuymaada Olympiad, 1

Tags: number theory

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

2012 Estonia Team Selection Test, 2

Tags: number theory , sequence

For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold: (1) $a_i = a_{i+n}$ for any $i$, (2) $a_i$ is not divisible by $n$ for any $i$, (3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$. For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?

2016 Iran Team Selection Test, 6

Tags: function , number theory , greatest common divisor

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

2013 Irish Math Olympiad, 1

Tags: exponent , number theory

Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.

2023 Junior Balkan Team Selection Tests - Moldova, 3

Tags: number theory

Prove that the number $A=2024^{n+1}-2023n-2024$ has at least $15$ different positive divisors for every nonnegative integer $ n $.

2023 Chile TST Ibero., 3

Tags: number theory

Determine the smallest positive integer $ n $ with the following property: for every triple of positive integers $ x, y, z $, with $ x $ dividing $ y^3 $, $ y $ dividing $ z^3 $, and $ z $ dividing $ x^3 $, it also holds that $ (xyz) $ divides $ (x + y + z)^n $.

2005 Chile National Olympiad, 5

Tags: number theory , functional

Compute $g(2005)$ where $g$ is a function defined on the natural numbers that has the following properties: i) $g(1) = 0$ ii) $g(nm) = g(n) + g(m) + g(n)g(m)$ for any pair of integers $n, m$. iii) $g(n^2 + 1) = (g(n) + 1)^2$ for every integer $n$.

2015 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , equation , algebra , diophantine equation

Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers

1995 Portugal MO, 1

Tags: number theory

Joao Salta-Pocinhas jumps $1$ meter in the first jump, $2$ meters in the second, $4$ meters in the third, . . ., $2^{n-1}$ meters in jump number $n$. Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?

1981 Tournament Of Towns, (013) 3

Tags: number theory , decimal representation , sum

Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$. (E Turkevich)

2022 IMAR Test, 1

Tags: number theory

Find all pairs of primes $p, q<2023$ such that $p \mid q^2+8$ and $q \mid p^2+8$.

1984 Austrian-Polish Competition, 2

Tags: gcd , greatest common divisor , number theory

Let $A$ be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of $n\in A$ yields a number $f (n) \in A$ (for instance, $f (3111) = 1333$). Find those $n \in A$ with $n > f (n)$ for which $gcd(n, f (n))$ is the largest possible.

2016 Nordic, 1

Tags: number theory , algebra

Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$.

2001 Paraguay Mathematical Olympiad, 3

Tags: number theory , sum of digits , divisible

Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.

2015 Estonia Team Selection Test, 3

Tags: number theory , rational

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

2014 IFYM, Sozopol, 5

Tags: polynomial , number theory , coprime

Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.

2021 Indonesia TST, N

Tags: number theory

A positive integer $n$ is said to be $interesting$ if there exist some coprime positive integers $a$ and $b$ such that $n = a^2 - ab + b^2$. Show that if $n^2$ is $interesting$, then $n$ or $3n$ is $interesting$.