Olympiad problems

1998 USAMO, 1

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \] ends in the digit $9$.

2011 NZMOC Camp Selection Problems, 6

Tags: diophantine equation , diophantine , number theory

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

2018 Canadian Mathematical Olympiad Qualification, 7

Tags: number theory , prime factorization , algebra , polynomial

Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$. Find all polynomials $P(x)$ with rational coefficients such that there exists infinitely many positive integers $n$ with $P(n) = rad(n)$.

JOM 2015 Shortlist, N1

Tags: number theory

Prove that there exists an infinite sequence of positive integers $ a_1, a_2, ... $ such that for all positive integers $ i $, \\ i) $ a_{i + 1} $ is divisible by $ a_{i} $.\\ ii) $ a_i $ is not divisible by $ 3 $.\\ iii) $ a_i $ is divisible by $ 2^{i + 2} $ but not $ 2^{i + 3} $.\\ iv) $ 6a_i + 1 $ is a prime power.\\ v) $ a_i $ can be written as the sum of the two perfect squares.

2014 CHKMO, 3

Tags: number theory , diophantine equation

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

2016 IMAR Test, 4

Tags: number theory , perfect number

A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.

2012 Argentina National Olympiad, 4

Tags: perfect square , number theory

For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

1982 Tournament Of Towns, (025) 3

Tags: number theory , diophantine , diophantine equation , factorial

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

2023 Bulgarian Autumn Math Competition, 11.3

Tags: number theory

Find the smallest possible number of divisors a positive integer $n$ may have, which satisfies the following conditions: 1. $24 \mid n+1$; 2. The sum of the squares of all divisors of $n$ is divisible by $48$ ($1$ and $n$ are included).

2022-23 IOQM India, 18

Tags: number theory

Let $m,n$ be natural numbers such that \\ $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$\\ Find the maximum possible value of $m+n$.

2011 Purple Comet Problems, 20

Tags: probability , perimeter , ratio , number theory , relatively prime , geometry

Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\tfrac{17}{128}$. There are relatively prime positive integers m and n so that the ratio of $DE$ to $AB$ is $\tfrac{m}{n}.$ Find $m + n$.

1990 USAMO, 3

Tags: modular arithmetic , number theory , greatest common divisor , relatively prime , combinatorics

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)

2013 Mexico National Olympiad, 1

Tags: inequalities , prime number , number theory

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

2025 Ukraine National Mathematical Olympiad, 8.3

Tags: number theory

Initially, there are $14$ numbers written on the board - zeros and ones. Every minute, Anton chooses half of the numbers on the board and adds $1$ to each of them, while Mykhailo multiplies all the other numbers by $8$. At some point (possibly initially), all the numbers on the board become equal. How many ones could have been on the board initially? [i]Proposed by Oleksii Masalitin[/i]

2023 CUBRMC, 8

Tags: number theory

If $r$ is real number sampled at random with uniform probability, find the probability that $r$ is [i]strictly [/i] closer to a multiple of $58$ than it is to a multiple of $37$.

2021 Science ON Seniors, 2

Tags: number theory , divisibility , order

Find all pairs $(p,q)$ of prime numbers such that $$p^q-4~|~q^p-1.$$ [i](Vlad Robu)[/i]

2023 HMNT, 5

Tags: number theory

Compute the unique positive integer $n$ such that $\frac{n^3-1989}{n}$ is a perfect square.

2010 Contests, 2

Tags: number theory

Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.

2023 Brazil National Olympiad, 4

Tags: number theory , diophantine equation , minimum value , algebra

Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$

2013 Romania Team Selection Test, 1

Tags: inequalities , algebra , polynomial , absolute value , triangle inequality , number theory

Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that \[ \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\] for every positive integer $n$.

1994 Portugal MO, 1

Tags: number theory

Determine the smallest natural number that has exactly $1994$ divisors.

2008 Junior Balkan Team Selection Tests - Romania, 1

Tags: induction , modular arithmetic , number theory , logarithm

Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.

2019 Turkey Team SeIection Test, 8

Tags: number theory

Let $p>2$ be a prime number, $m>1$ and $n$ be positive integers such that $\frac {m^{pn}-1}{m^n-1}$ is a prime number. Show that: $$pn\mid (p-1)^n+1$$

1987 Yugoslav Team Selection Test, Problem 1

Tags: recurrence relation , number theory , sequence

Let $x_0=a,x_1=b$ and $x_{n+1}=2x_n-9x_{n-1}$ for each $n\in\mathbb N$, where $a,b$ are integers. Find the necessary and sufficient condition on $a$ and $b$ for the existence of an $x_n$ which is a multiple of $7$.

2021 Taiwan Mathematics Olympiad, 2.

Tags: quadratic , arithmetic sequence , number theory

Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that \[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\] [i]Proposed by usjl[/i]