Olympiad problems

2017 Romania National Olympiad, 4

Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

2021 Brazil EGMO TST, 1

Tags: algebra , number theory

Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true: I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$ II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$ III- $x_0=1$ a) Determine the value of $x_{2022}$ b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$

2019 Iran MO (3rd Round), 2

Tags: number theory

Prove that for any positive integers $m>n$, there is infinitely many positive integers $a,b$ such that set of prime divisors of $a^m+b^n$ is equal to set of prime divisors of $a^{2019}+b^{1398}$.

2024 Dutch IMO TST, 1

Tags: divisor , coprime , number theory

For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2004 Federal Math Competition of S&M, 1

Tags: number theory

Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.

2021 Belarusian National Olympiad, 9.7

Tags: number theory

It is known that $(x-y)^3 \vdots 6x^2-2y^2$, where $x,y$ are some integers. Prove that then also $(x+y)^3 \vdots 6x^2-2y^2$.

2004 Iran Team Selection Test, 2

Tags: number theory , modular arithmetic

Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.

2010 Contests, 1

Tags: modular arithmetic , number theory

Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

2015 IFYM, Sozopol, 2

Tags: sequence , number theory

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

2006 Junior Tuymaada Olympiad, 6

Tags: sum , combinatorics , number theory

[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $. For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$. Find the number of all [i]palindromic partitions[/i] of the number $2006$.

KoMaL A Problems 2019/2020, A. 778

Tags: diophantine equation , number theory

Find all square-free integers $d$ for which there exist positive integers $x, y$ and $n$ satisfying $x^2+dy^2=2^n$ Submitted by Kada Williams, Cambridge

2006 Bosnia and Herzegovina Team Selection Test, 3

Tags: positive integer , fractional part , inequality , inequalities , number theory

Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.

2018 Pan-African Shortlist, N3

Tags: sequence , algebra , number theory

For any positive integer $x$, we set $$ g(x) = \text{ largest odd divisor of } x, $$ $$ f(x) = \begin{cases} \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\ 2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.} \end{cases} $$ Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.

2000 Taiwan National Olympiad, 1

Tags: euler , quadratic , prime number , number theory , modular arithmetic , function

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

PEN H Problems, 77

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

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2000 All-Russian Olympiad Regional Round, 9.2

Tags: divide , divisible , number theory

Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?

2020 HMIC, 4

Tags: catalan , polynomial , number theory , algebra

Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set \[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\] are divisible by $p$. [i]Tristan Shin[/i]

2014 BMT Spring, 3

Tags: number theory

The Professor chooses to assign homework problems from a set of problems labeled $1$ to $100$, inclusive. He will not assign two problems whose numbers share a common factor greater than $1$. If the Professor chooses to assign the maximum number of homework problems possible, how many different combinations of problems can he assign?

2006 India IMO Training Camp, 2

Tags: combinatorics , induction , number theory

Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.

2006 Hungary-Israel Binational, 1

Tags: induction , modular arithmetic , number theory

If natural numbers $ x$, $ y$, $ p$, $ n$, $ k$ with $ n > 1$ odd and $ p$ an odd prime satisfy $ x^n \plus{} y^n \equal{} p^k$, prove that $ n$ is a power of $ p$.

2017 USA TSTST, 6

Tags: number theory , fibonacci , algebra

A sequence of positive integers $(a_n)_{n \ge 1}$ is of [i]Fibonacci type[/i] if it satisfies the recursive relation $a_{n + 2} = a_{n + 1} + a_n$ for all $n \ge 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences? [i]Proposed by Ivan Borsenco[/i]

2018 Danube Mathematical Competition, 3

Tags: positive integer , sum , product , rational , number theory

Find all the positive integers $n$ with the property: there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$ such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.

2017 IFYM, Sozopol, 4

Tags: number theory

Prove that, if there exist natural numbers $a_1,a_2…a_{2017}$ for which the product $(a_1^{2017}+a_2 )(a_2^{2017}+a_3 )…(a_{2016}^{2017}+a_{2017})(a_{2017}^{2017}+a_1)$ is a $k$-th power of a prime number, then $k=2017$ or $k\geq 2017.2018$.

2022 Estonia Team Selection Test, 5

Tags: number theory

(a) Is it true that, for arbitrary integer $n{}$ greater than $1$ and distinct positive integers $i{}$ and $j$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'}$ and $j^{'}$ whose product $i^{'}j^{'}$ is divisible by the product $ij$? (b) Is it true that, for arbitrary integer $n{}$ greater than $2$ and distinct positive integers $i, j, k$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'},j^{'},k^{'}$ whose product $i^{'}j^{'}k^{'}$ is divisible by the product $ijk$?

2020 Federal Competition For Advanced Students, P2, 3

Tags: prime number , sequence , number theory

Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and $$ e_n=a + \prod_{k=0}^{n-1} e_k$$ for $n \geq 1$. Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence. (Theresia Eisenkölbl)