Olympiad problems

2012 ELMO Shortlist, 5

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]

1994 Baltic Way, 9

Tags: modular arithmetic , number theory , search

Find all pairs of positive integers $(a,b)$ such that $2^a+3^b$ is the square of an integer.

2005 Austrian-Polish Competition, 4

Tags: number theory , perfect square , fermat's little theorem , prime , diophantine equation

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

2023 ELMO Shortlist, N4

Tags: number theory

Let $d(n)$ denote the number of positive divisors of $n$. The sequence $a_0$, $a_1$, $a_2$, $\ldots$ is defined as follows: $a_0=1$, and for all integers $n\ge1$, \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers $n\ge1$, we have $a_n\le3n$. [i]Proposed by Karthik Vedula[/i]

1997 Belarusian National Olympiad, 2

Tags: number theory , algebra , recurrence relation , number theory with sequences

A sequence $(a_n)_{-\infty}^{-\infty}$ of zeros and ones is given. It is known that $a_n = 0$ if and only if $a_{n-6} + a_{n-5} +...+ a_{n-1}$ is a multiple of $3$, and not all terms of the sequence are zero. Determine the maximum possible number of zeros among $a_0,a_1,...,a_{97}$.

Kvant 2020, M2597

Tags: prime number , number theory

Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$. [i]Proposed by M. Antipov[/i]

2000 AIME Problems, 12

Tags: number theory , geometry , trigonometry , circumcircle , 3d geometry , sphere , tetrahedron

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

2018 Malaysia National Olympiad, A2

Tags: number theory , digit

An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

1978 IMO Longlists, 22

Tags: number theory , special factorizations

Let $x$ and $y$ be two integers not equal to $0$ such that $x+y$ is a divisor of $x^2+y^2$. And let $\frac{x^2+y^2}{x+y}$ be a divisor of $1978$. Prove that $x = y$. [i]German IMO Selection Test 1979, problem 2[/i]

2010 Contests, 3

Tags: number theory

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

2012 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , combinatorics

Positive integers $a, b, c$ have greatest common divisor $1$. The triplet $(a, b, c)$ may be altered into another triplet such that in each step one of the numbers in the actual triplet is increased or decreased by an integer multiple of another element of the triplet. Prove that the triplet $(1,0,0)$ can be obtained in at most $5$ steps.

1972 IMO Longlists, 41

Tags: number theory

The ternary expansion $x = 0.10101010\cdots$ is given. Give the binary expansion of $x$. Alternatively, transform the binary expansion $y = 0.110110110 \cdots$ into a ternary expansion.

1998 VJIMC, Problem 1

Tags: number theory , arithmetic progression

Let $a$ and $d$ be two positive integers. Prove that there exists a constant $K$ such that every set of $K$ consecutive elements of the arithmetic progression $\{a+nd\}_{n=1}^\infty$ contains at least one number which is not prime.

2024 Korea Summer Program Practice Test, 8

Tags: number theory

For a positive integer $ n $, let $ \tau(n) $ denote the number of positive divisors of $ n $. Determine whether there exists a positive integer triple $ a, b, c $ such that there are exactly $1012$ positive integers $ K $ not greater than $2024$ that satisfies the following: the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \] holds for some positive integers $x,y,z$.

2005 ITAMO, 2

Tags: number theory , modular arithmetic , function , algebra , euler , totient function

Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and \[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\] For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?

2012 ITAMO, 2

Tags: modular arithmetic , number theory

Determine all positive integers that are equal to $300$ times the sum of their digits.

2021 OMpD, 3

Tags: number theory , diophantine equation

Determine all pairs of integer numbers $(x, y)$ such that: $$\frac{(x - y)^2}{x + y} = x - y + 6$$

2016 Saudi Arabia IMO TST, 3

Tags: number theory , divisible , combinatorics

Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that: $\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$. $\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$. Find the maximum value of $n$

2008 Postal Coaching, 3

Tags: number theory , divide , divisible

Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.

2010 Bosnia Herzegovina Team Selection Test, 1

Tags: prime number , number theory

$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

2013 Serbia Additional Team Selection Test, 1

Tags: number theory

We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$ ($a_n b_m \neq 0$) similar if the following conditions hold: $(i)$ $n = m$; $(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$. Let $P(x)$ and $Q(x)$ be similar polynomials with integer coefficients. Given that $P(16) = 3^{2012}$, find the smallest possible value of $|Q(3^{2012})|$. [i]Proposed by Milos Milosavljevic[/i]

1990 IMO, 1

Tags: number theory , algebra , function , functional equation

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

2011 Iran Team Selection Test, 2

Tags: least common multiple , arithmetic sequence , number theory

Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.

1996 May Olympiad, 2

Tags: number theory , digit

Considering the three-digit natural numbers, how many of them, when adding two of their digits, are double of their remainder? Justify your answer.

1990 Austrian-Polish Competition, 2

Tags: number theory , diophantine , diophantine equation

Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.