Found problems: 295
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.
Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
A set $S$ of natural numbers is called [i]good[/i], if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a [i]good [/i]set which is a subset of the set $A = \{1,2, 3, ...,63\}$.
A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$.
[i]Proposed by Michał Pilipczuk[/i]
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
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[*] $A\cap B=\{1\};$
[*] 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.
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Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.
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 $C$.
Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^2, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
Is there a countable set $Y$ and an uncountable family $\mathcal F$ of its subsets such that for every two distinct $A,B\in\mathcal F$, their intersection $A\cap B$ is finite?
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 $A$?
Find the minimum positive integer $n$ such that for any set $A$ with $n$ positive intergers has $15$ elements which sum is divisible by $15$.
Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.
A positive integer $n$ is given. If there exist sets $F_1, F_2, \cdots F_m$ satisfying the following, prove that $m \le n$.
(For sets $A, B$, $|A|$ is the number of elements in $A$. $A-B$ is the set of elements that are in $A$ but not $B$)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots n\}$
(ii): $|F_1| \le |F_2| \le \cdots \le |F_m|$
(iii): For all $1 \le i < j \le m$, $|F_i-F_j|=1$.
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$.
a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying:
[b]i.[/b] each has $k$ elements;
[b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$)
[b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements.
b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.
2. Let $A_1,A_2,...,A_n$be distinct subsets of an n-element set $ X$ ($n \geq 2$). Show that there exists an element $x$ of $X$ such that the sets $A_1\setminus \{x\}$ ,:......., $A_n\setminus \{x\}$ are all distinct.
Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$. The elements of $A$ are colored in such a way that:
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[*] each element of $P$ has a different color,
[*] if $m,n \in A$, then $mn$ is the same color of $m$ or $n$,
[*] for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$, there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$, so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$, simultaneously.
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Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.
Let $m,n$ be positive integers and let $[a]$ denote the residue class$\pmod{mn}$ of the integer $a$ (thus $\{[r]|r\text{ is an integer}\}$ has exactly $mn$ elements). Suppose the set $\{[ar]|r\text{ is an integer}\}$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is coprime to $mn$ and $[nq]=[a]$.
Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\ldots,m$ satisfying the following conditions:
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[*]$B_i\subset A,$ $|B_i|=k,$
[*]$C_i\subset B_i$ (there is no additional condition for the number of elements in $C_i$), and
[*]$B_i\cap C_j\neq B_j\cap C_i$ for all $i\neq j.$
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Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$.
(a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element.
(b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
(c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
There is a set of $17$ consecutive positive integers. Let $m$ be the smallest of these numbers. Determine for which values of $m$ the set can be divided into three subsets disjoint, such that the sum of the elements of each subset is the same.