Olympiad problems

2022 Germany Team Selection Test, 1

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

2002 Romania Team Selection Test, 1

Tags: number theory

Let $m,n$ be positive integers of distinct parities and such that $m<n<5m$. Show that there exists a partition with two element subsets of the set $\{ 1,2,3,\ldots ,4mn\}$ such that the sum of numbers in each set is a perfect square. [i]Dinu Șerbănescu[/i]

1998 Harvard-MIT Mathematics Tournament, 5

Tags: floor function , number theory , relatively prime , complementary counting

How many positive integers less than $1998$ are relatively prime to $1547$? (Two integers are relatively prime if they have no common factors besides 1.)

2021-IMOC, N5

Tags: number theory , greatest common divisor , set

Find all sets $S$ of positive integers that satisfy all of the following. $1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$. $2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$. $3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.

2022 Baltic Way, 17

Tags: number theory

Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$. Prove that $n$ has at least $2022$ distinct prime factors.

2016 China National Olympiad, 4

Tags: number theory

Let $n \geq 2$ be a positive integer and define $k$ to be the number of primes $\leq n$. Let $A$ be a subset of $S = \{2,...,n\}$ such that $|A| \leq k$ and no two elements in $A$ divide each other. Show that one can find a set $B$ such that $|B| = k$, $A \subseteq B \subseteq S$ and no two elements in $B$ divide each other.

2015 Thailand Mathematical Olympiad, 8

Tags: perfect square , odd , number theory

Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

1998 IMO, 4

Tags: number theory , divisibility , polynomial

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

2014 USAMTS Problems, 2:

Tags: modular arithmetic , search , number theory , prime number

Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.

2019 Poland - Second Round, 3

Tags: rational number , algebra , number theory , function

Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}

2018 Estonia Team Selection Test, 6

Tags: number theory , digit

We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

2012 China Girls Math Olympiad, 7

Tags: search , analytic geometry , number theory

Let $\{a_n\}$ be a sequence of nondecreasing positive integers such that $\textstyle\frac{r}{a_r} = k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that $\textstyle\frac{s}{a_s} = k$.

2015 Saudi Arabia BMO TST, 4

Tags: divide , divisible , number theory

Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

2021 Thailand TST, 1

Tags: number theory , sequence , divisibility

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

1986 Federal Competition For Advanced Students, P2, 4

Tags: number theory

Find the largest $ n$ for which there is a natural number $ N$ with $ n$ decimal digits which are all different such that $ n!$ divides $ N$. Furthermore, for this largest $ n$ find all possible numbers $ N$.

2014 NIMO Problems, 3

Tags: trigonometry , analytic geometry , symmetry , geometry , number theory , relatively prime , similar triangles

Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$? [i]Proposed by David Altizio[/i]

2016 PUMaC Individual Finals A, 2

Tags: floor function , number theory

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

Russian TST 2018, P3

Tags: number theory , euclidean algorithm

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties: [list] [*] $f(1,1)=0$. [*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1; [*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$. [/list] Prove that $$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$

2006 India Regional Mathematical Olympiad, 6

Tags: search , number theory

Prove that there are infinitely many positive integers $ n$ such that $ n(n\plus{}1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $ a^{2}\plus{}b^{2}$ and $ b^{2}\plus{}a^{2}$ are considered as the same representation.)

2023 LMT Spring, 10

Tags: number theory

Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.

2001 Korea Junior Math Olympiad, 2

Tags: number theory , relatively prime , algebra , polynomial

$n$ is a product of some two consecutive primes. $s(n)$ denotes the sum of the divisors of $n$ and $p(n)$ denotes the number of relatively prime positive integers not exceeding $n$. Express $s(n)p(n)$ as a polynomial of $n$.

1983 All Soviet Union Mathematical Olympiad, 364

Tags: number theory , combinatorics

The kindergarten group is standing in the column of pairs. The number of boys equals the number of girls in each of the two columns. The number of mixed (boy and girl) pairs equals to the number of the rest pairs. Prove that the total number of children in the group is divisible by eight.

2020 BMT Fall, 26

Tags: number theory

Estimate the value of the $2020$th prime number $p$ such that $p + 2$ is also prime. If $E > 0$ is your estimate and $A$ is the correct answer, you will receive $25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^2$ points, rounded to the nearest integer. (An estimate less than or equal to $0$ will receive $0$ points.

2000 JBMO ShortLists, 4

Tags: number theory

Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$.

2023 Regional Olympiad of Mexico West, 1

Tags: number theory , divisor

For every positive integer $n$ we take the greatest divisor $d$ of $n$ such that $d\leq \sqrt{n}$ and we define $a_n=\frac{n}{d}-d$. Prove that in the sequence $a_1,a_2,a_3,...$, any non negative integer $k$ its in the sequence infinitely many times.