Olympiad problems

2014 JBMO TST - Macedonia, 5

Tags: combinatorics , set

Prove that there exist infinitely many pairwisely disjoint sets $A(1), A(2),...,A(2014)$ which are not empty, whose union is the set of positive integers and which satisfy the following condition: For arbitrary positive integers $a$ and $b$, at least two of the numbers $a$, $b$ and $GCD(a,b)$ belong to one of the sets $A(1), A(2),...,A(2014)$.

2014 IFYM, Sozopol, 6

Tags: combinatorics , relation , set

Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called [i]"harmonious"[/i], if they are not coprime. It is known that each element from $A$ is not [i]harmonious[/i] with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are [i]harmonious[/i] but $(a_1,b_2)$ and $(a_2,b_1)$ are not.

1996 Yugoslav Team Selection Test, Problem 1

Tags: number theory , set , combinatorics

Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions: (a) Any two distinct sets from $\mathfrak F$ have exactly one element in common; (b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$. Can $n$ be equal to $1996$?

2018 Junior Regional Olympiad - FBH, 3

Tags: activities , counting , set

In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of them also goes to spanish language. $10$ students who study spanish also study german. $12$ students who study german also do sports. Only $4$ students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?

2001 VJIMC, Problem 1

Tags: number theory , inequalities , set

Let $A$ be a set of positive integers such that for any $x,y\in A$, $$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.

2016 Korea Summer Program Practice Test, 5

Tags: combinatorics , set

Find the maximal possible $n$, where $A_1, \dots, A_n \subseteq \{1, 2, \dots, 2016\}$ satisfy the following properties. - For each $1 \le i \le n$, $\lvert A_i \rvert = 4$. - For each $1 \le i < j \le n$, $\lvert A_i \cap A_j \rvert$ is even.

2014 Saudi Arabia GMO TST, 4

Tags: rational , set , algebra

Let $X$ be a set of rational numbers satisfying the following two conditions: (a) The set $X$ contains at least two elements, (b) For any $x, y$ in $X$, if $x \ne y$ then there exists $z$ in $X$ such that either $\left| \frac{x- z}{y - z} \right|= 2$ or $\left| \frac{y -z}{x - z} \right|= 2$ . Prove that $X$ contains infinitely many elements.

2018 Brazil Undergrad MO, 4

Tags: set

Consider the property that each a element of a group $G$ satisfies $a ^ 2 = e$, where e is the identity element of the group. Which of the following statements is not always valid for a group $G$ with this property? (a) $G$ is commutative (b) $G$ has infinite or even order (c) $G$ is Noetherian (d) $G$ is vector space over $\mathbb{Z}_2$

2011 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , set

Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$ $a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$ $b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: arithmetic mean , number theory , set , combinatorics

Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$

2007 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: combinatorics , set

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets . Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .

1992 Romania Team Selection Test, 4

Tags: sequence , combinatorics , algebra , set

Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$

2017 Romanian Master of Mathematics Shortlist, A1

Tags: algebra , binary operation , set , cardinality

A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions: (1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ , (2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ , (3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A$, and (4) If a and b are distinct elements of $A-\{e\}$, then $a^3 * b = b^3 * a^2$, where $x^k = x * x^{k-1}$ for all integers $k \ge 2$ and all $x$ in $A$. Determine the largest cardinality $A$ may have. proposed by Bojan Basic, Serbia

2021 Switzerland - Final Round, 3

Tags: number theory , set

Find all finite sets $S$ of positive integers with at least $2$ elements, such that if $m>n$ are two elements of $S$, then $$ \frac{n^2}{m-n} $$ is also an element of $S$.

2001 Moldova National Olympiad, Problem 7

Tags: set , number theory

Let $n$ be a positive integer. We denote by $S$ the sum of elements of the set $M=\{x\in\mathbb N|(n-1)^2\le x<(n+1)^2\}$. (a) Show that $S$ is divisible by $6$. (b) Find all $n\in\mathbb N$ for which $S+(1-n)(1+n)=2001$.

2021 Korea - Final Round, P4

Tags: easy , set , combinatorics

There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below. (1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$ (2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$ (3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$

1999 Junior Balkan Team Selection Tests - Moldova, 5

Tags: algebra , set , irreducible , fraction

Let the set $M =\{\frac{1998}{1999},\frac{1999}{2000} \}$. The set $M$ is completed with new fractions according to the rule: take two distinct fractions$ \frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ from $M$ thus there are no other numbers in $M$ located between them and a new fraction is formed, $\frac{p_1+p_2}{q_1+q_2}$ which is included in $M$, etc. Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in $M$.

2011 VTRMC, Problem 6

Tags: set

Let $S$ be a set with an asymmetric relation $<$; this means that if $a,b\in S$ and $a<b$, then we do not have $b<a$. Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a,b\in S$, then $a<b$ if and only if $a\prec b$, and if $x,y\in T$ with $x\prec y$, then there exists $t\in T$ such that $x\prec t\prec y$.

1974 Poland - Second Round, 1

Tags: combinatorics , subset , set

Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.

2021 Junior Balkan Team Selection Tests - Romania, P3

Tags: number theory , maximum value , set

Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$ The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.

1984 Putnam, B3

Tags: abstract algebra , set

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$, $(\text i)$ $x*z=y*z$ implies $x=y$ $(\text{ii})$ $x*(y*z)\ne(x*y)*z$

2021 Science ON grade VII, 1

Tags: number theory , set

Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

2009 Junior Balkan Team Selection Tests - Romania, 3

Tags: product , set , number theory

Let $A$ be a finite set of positive real numbers satisfying the property: [i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i] Show that the product of all elements in $A$ is equal to $1$.

2017 Azerbaijan EGMO TST, 1

Tags: set , number theory

$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?

2015 NIMO Summer Contest, 6

Tags: combinatorics , set

Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set \[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \] [i] Proposed by Evan Chen [/i]