Olympiad problems

1997 All-Russian Olympiad, 1

Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. [i]M. Sonkin[/i]

2010 AIME Problems, 2

Tags: probability , number theory , relatively prime

A point $ P$ is chosen at random in the interior of a unit square $ S$. Let $ d(P)$ denote the distance from $ P$ to the closest side of $ S$. The probability that $ \frac15\le d(P)\le\frac13$ is equal to $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

2004 Spain Mathematical Olympiad, Problem 6

Tags: circles , pattern , number theory

We put, forming a circumference of a circle, ${2004}$ bicolored files: white on one side of the file and black on the other. A movement consists in choosing a file with the black side upwards and flipping three files: the one chosen, the one to its right, and the one to its left. Suppose that initially there was only one file with its black side upwards. Is it possible, repeating the movement previously described, to get all of the files to have their white sides upwards? And if we were to have ${2003}$ files, between which exactly one file began with the black side upwards?

2017 OMMock - Mexico National Olympiad Mock Exam, 3

Tags: square , number theory , system

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

2018 Auckland Mathematical Olympiad, 2

Tags: number theory , digit

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

2013 India PRMO, 13

Tags: number theory , maximum , divide

To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?

2005 Korea Junior Math Olympiad, 6

Tags: set theory , prime , number theory

For two different prime numbers $p, q$, define $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.

2019 PUMaC Algebra B, 4

Tags: algebra , number theory , relatively prime

Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.

2022 Junior Balkan Team Selection Tests - Moldova, 11

Tags: divide , number theory

Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.

2015 Baltic Way, 20

Tags: number theory

For any integer $n \ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \ge 2 $ for which $ A_n$ is odd. (Note: The distance between two integers $ a$ and $b$ is defined as $|a -b|$.)

2017 Korea Winter Program Practice Test, 4

Tags: combinatorics , number theory

For a nonzero integer $k$, denote by $\nu_2(k)$ the maximal nonnegative integer $t$ such that $2^t \mid k$. Given are $n (\ge 2)$ pairwise distinct integers $a_1, a_2, \ldots, a_n$. Show that there exists an integer $x$, distinct from $a_1, \ldots, a_n$, such that among $\nu_2(x - a_1), \ldots, \nu_2(x - a_n)$ there are at least $n/4$ odd numbers and at least $n/4$ even numbers.

2021 Philippine MO, 7

Tags: algebra , number theory , inequalities , philippines

Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$ $$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$.

1991 ITAMO, 2

Tags: perfect square , number theory

Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.

2007 Kyiv Mathematical Festival, 1

Tags: number theory

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

2014 ELMO Shortlist, 8

Tags: function , greatest common divisor , number theory

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

1990 Polish MO Finals, 3

Tags: modular arithmetic , number theory

Prove that for all integers $n > 2$, \[ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} \]

2020 HMNT (HMMO), 5

Tags: number theory

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $[\frac{a}{b}, \frac{c}{d} )$ where $a,b,c,d$ are positive integers with $\gcd(a,b) = \gcd(c,d) = 1$. Compute $1000a+100b+10c+d$.

2019 CMIMC, 5

Tags: number theory

Let $x_n$ be the smallest positive integer such that $7^n$ divides $x_n^2-2$. Find $x_1+x_2+x_3$.

2017 Saudi Arabia JBMO TST, 6

Tags: number theory , perfect square

Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.

1992 Vietnam Team Selection Test, 1

Tags: floor function , number theory

Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.

2003 Peru Cono Sur TST, P2

Tags: number theory

Let $p$ and $n$ be positive integers such that $p$ is prime and $1 + np$ is a perfect square. Prove that the number $n + 1$ can be expressed as the sum of $p$ perfect squares, where some of them can be equal.

2015 India PRMO, 15

Tags: number theory , digit , prime number

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

2011 China Girls Math Olympiad, 1

Tags: diophantine equation , prime factorization , number theory

Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$.

2010 IMO Shortlist, 4

Tags: modular arithmetic , divisibility , number theory

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2010 Indonesia TST, 3

Tags: number theory

Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]