Olympiad problems

2007 Thailand Mathematical Olympiad, 15

Compute the remainder when $222!^{111} + 111^{222!} + 111!^{222} + 222^{111!}$ is divided by $2007$.

2023 Turkey Olympic Revenge, 4

Tags: number theory , functional equation , revenge

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$ is a perfect square. [i]Proposed by Barış Koyuncu[/i]

1916 Eotvos Mathematical Competition, 3

Tags: combinatorics , number theory

Divide the numbers $$1, 2,3, 4,5$$ into two arbitrarily chosen sets. Prove that one of the sets contains two numbers and their difference.

2014 Middle European Mathematical Olympiad, 7

Tags: least common multiple , number theory

A finite set of positive integers $A$ is called [i]meanly[/i] if for each of its nonempy subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if $\frac{1}{k}(a_1 + \dots + a_k)$ is an integer whenever $k \ge 1$ and $a_1, \dots, a_k \in A$ are distinct. Given a positive integer $n$, determine the least possible sum of the elements of a meanly $n$-element set.

2020 China Northern MO, BP5

Tags: combinatorics , set , number theory , relatively prime

It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition $$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$ Determine the maximum positive integer $n$. *$(a,b)$ means $\gcd (a,b)$

2024 IRN-SGP-TWN Friendly Math Competition, 4

Tags: function , number theory

Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying \[f_k(x)=x+k\varphi(x)\] where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with [list] [*] $a_1=c$, and [*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$ [/list] Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if (a) $k=2025$ ? (b) $k=2065$ ? [i]Proposed by chorn[/i]

2016 Baltic Way, 3

Tags: diophantine equation , number theory

For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$ have a solution in integers?

2008 ITest, 39

Tags: number theory , function , relatively prime

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

2025 Taiwan TST Round 2, N

Tags: number theory

Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and \[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\] for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.

2009 Harvard-MIT Mathematics Tournament, 4

Tags: algebra , least common multiple , greatest common divisor , quadratic , number theory , polynomial

Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.

2015 Costa Rica - Final Round, 4

Tags: number theory , integer

Find all triples of integers $(x, y, z)$ not zero and relative primes in pairs such that $\frac{(y+z-x)^2}{4x}$, $\frac{(z+x-y)^2}{4y}$ and $\frac{(x+y-z)^2}{4z}$ are all integers.

2022 239 Open Mathematical Olympiad, 4

Tags: prime , natural number , number theory

Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.

2023 Tuymaada Olympiad, 1

Tags: number theory , combinatorics

The numbers $1, 2, 3, \ldots$ are arranged in a spiral in the vertices of an infinite square grid (see figure). Then in the centre of each square the sum of the numbers in its vertices is placed. Prove that for each positive integer n the centres of the squares contain infinitely many multiples of $n$.

2022 Canada National Olympiad, 2

Tags: number theory

I think we are allowed to discuss since its after 24 hours How do you do this Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function

2018 International Zhautykov Olympiad, 3

Tags: number theory

Prove that there exist infinitely pairs $(m,n)$ such that $m+n$ divides $(m!)^n+(n!)^m+1$

2018 IFYM, Sozopol, 8

Tags: number theory

Are there infinitely many positive integers that [b]can’t[/b] be presented as a sum of no more than fifteen fourth degrees of positive integers. (For example 15 isn’t such number as it can be presented as the sum of $15.1^4$)

2011 CentroAmerican, 4

Tags: prime number , number theory

Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$

1969 Yugoslav Team Selection Test, Problem 4

Tags: number theory

Let $a$ and $b$ be two natural numbers such that $a<b$. Prove that in each set of $b$ consecutive positive integers there are two numbers whose product is divisible by $ab$.

2022 Turkey Team Selection Test, 1

Tags: modular arithmetic , diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

1985 Tournament Of Towns, (091) T2

Tags: combinatorics , number theory , difference , prime , composite , maximum

From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .

1997 Greece National Olympiad, 3

Tags: number theory

Find all integer solutions to \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}.\]

2011 China Western Mathematical Olympiad, 4

Tags: number theory

Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$

2022 China Girls Math Olympiad, 4

Tags: number theory

Given a prime number $p\ge 5$. Find the number of distinct remainders modulus $p$ of the product of three consecutive positive integers.

1994 Mexico National Olympiad, 4

Tags: integer sequence , number theory , relatively prime

A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$. What is the last page to be read?

2018 USA Team Selection Test, 1

Tags: relatively prime , arithmetic function , number theory

Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds. [i]Proposed by Ashwin Sah[/i]