Olympiad problems

1989 Romania Team Selection Test, 2

Let $P$ be a point on a circle $C$ and let $\phi$ be a given angle incommensurable with $2\pi$. For each $n \in N, P_n$ denotes the image of $P$ under the rotation about the center $O$ of $C$ by the angle $\alpha_n = n \phi$. Prove that the set $M = \{P_n | n \ge 0\}$ is dense in $C$.

1989 Romania Team Selection Test, 4

Tags: function , divisor , combinatorics , set

A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that a) $f(4k +2) = 3$ for each nonnegative integer $k$, b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$

1962 Putnam, A6

Tags: rational number , set

Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $ab$ and $a+b$, and having the property that for every rational number $r$ exactly one of the following three statements is true: $$r\in S,\;\; -r\in S,\;\;r =0.$$ Prove that $S$ is the set of all positive rational numbers.

2020-2021 Winter SDPC, #7

Tags: algebra , set

Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.

2004 Federal Math Competition of S&M, 3

Tags: combinatorics , set

Let $A = \{1,2,3, . . . ,11\}$. How many subsets $B$ of $A$ are there, such that for each $n\in \{1,2, . . . ,8\}$, if $n$ and $n+2$ are in $B$ then at least one of the numbers $ n+1$ and $n+3$ is also in $B$?

2013 Danube Mathematical Competition, 4

Tags: set theory , combinatorics , set

Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$

2016 India PRMO, 14

Tags: minimum , subset , set

Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.

1985 All Soviet Union Mathematical Olympiad, 410

Tags: algebra , set , absolute value

Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$

2018 IMAR Test, 3

Tags: combinatorics , set

Let $S$ be a finite set and let $\mathcal{P}(S)$ be its power set, i.e., the set of all subsets of $S$, the empty set and $S$, inclusive. If $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S),$ let \[\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.\] Given a non-negative integer $n\leqslant |S|,$ determine the minimal size $\mathcal{A}\vee \mathcal{B}$ may have, where $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S)$ such that $|\mathcal{A}|+|\mathcal{B}|>2^n$. [i]Amer. Math. Monthly[/i]

2021 Science ON grade VII, 4

Tags: equality case , combinatorics , set

Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$. Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have. [i] (Andrei Bâra)[/i]

2013 VJIMC, Problem 3

Tags: polynomial , number theory , set

Let $S$ be a finite set of integers. Prove that there exists a number $c$ depending on $S$ such that for each non-constant polynomial $f$ with integer coefficients the number of integers $k$ satisfying $f(k)\in S$ does not exceed $\max(\deg f,c)$.

2024 Israel TST, P3

Tags: distance , combinatorial geometry , combinatorics , geometry , set

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

2016 India IMO Training Camp, 3

Tags: number theory , set , prime number

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

2017 Bosnia and Herzegovina Junior BMO TST, 2

Tags: combinatorics , subset , set

Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.

2013 German National Olympiad, 5

Tags: combinatorics , intersection , set

Five people form several commissions to prepare a competition. Here any commission must be nonempty and any two commissions cannot contain the same members. Moreover, any two commissions have at least one common member. There are already $14$ commissions. Prove that at least one additional commission can be formed.

2007 Nicolae Coculescu, 4

Tags: floor function , number theory , set

Let be a natural number $ n\ge 2. $ Prove that there exists an unique bipartition $ \left( A,B \right) $ of the set $ \{ 1,2\ldots ,n \} $ such that $ \lfloor \sqrt x \rfloor\neq y , $ for any $ x,y\in A , $ and $ \lfloor \sqrt z \rfloor\neq t , $ for any $ z,t\in B. $ [i]Costin Bădică[/i]

2018 China Girls Math Olympiad, 6

Tags: combinatorics , set , integer

Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have (i) $168 \in I_i$; (ii) $\forall x, y \in I_i$, we have $x-y \in I_i$. Determine the number of $k-chain$ in total.

2015 Indonesia MO Shortlist, C2

Tags: combinatorics , arithmetic , set

Given $2n$ natural numbers, so that the average arithmetic of those $2n$ number is $2$. If all the number is not more than $2n$. Prove we can divide those $2n$ numbers into $2$ sets, so that the sum of each set to be the same.

2014 Danube Mathematical Competition, 2

Tags: floor function , set , algebra , product

Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.

2014 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: remainder , number theory , combinatorics , set

Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions: $\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$. $\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$. $\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$. Note: One or more of the sets may be empty.

2013 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , set

If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$

2006 Lithuania National Olympiad, 4

Tags: subset , cardinality , uniqueness , combinatorics , set

Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b | a, b \in S, a \neq b$ are equal.

2009 BAMO, 3

Tags: number theory , greatest common divisor , set

A set $S$ of positive integers is called magic if for any two distinct members of $S, i$ and $j$, $\frac{i+ j}{GCD(i, j)}$is also a member of $S$. The $GCD$, or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, $GCD(36,80) = 4$. Find and describe all finite magic sets.

2019 India PRMO, 21

Tags: number theory , prime number , set , prime

Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

1996 Tuymaada Olympiad, 2

Tags: algebra , set theory , real , set

Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?