Found problems: 295
Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be [i]friends[/i] if the following conditions are true:
[list]
[*] They do not share any elements.
[*] They both have the same number of elements.
[*] The product of all elements from $A$ equals the product of all elements from $B$.
[/list]
Prove that there are two subsets of $S$ that are [i]friends[/i] such that each one of them contains at least $738$ elements.
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]
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$.
(Alberto Alfarano)
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 $nx + S =\{nx + s : s \in S\}$
Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have?
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]
Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$.
[i]Proposed by Viktor Simjanoski[/i]
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]
Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that:
every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number
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 $r + B = \{ r + b | b \in B\}$ .
Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them and gets the same set of numbers as in the beginning. What were the starting numbers?
Find all the finite sets $A$ of real positive numbers having at least two elements, with the property that $a^2 + b^2 \in A$ for every $a, b \in A$ with $a \ne b$
There are two points $A$ and $B$ in the plane.
a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$
b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.
Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.
Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}$ for $k = 2,3,4, ...$
(a) Determine $N - \cup _{k=1}^{\infty} S_k$.
(b) Find all $n$ for which $1994 \in S_n$.
In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct?
[list=1]
[*] Only 10 boys use spectacles
[*] Only 20 girls use spectacles
[*] At most 25 boys do not use spectacles
[*] At most 30 girls do not use spectacles
[/list]
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
We make groups of numbers. Each group consists of [i]fi ve[/i] distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups.
(a) Determine whether it is possible to make $2015$ groups.
(b) If all groups together must contain exactly [i]six [/i] distinct numbers, what is the greatest number of groups that you can make?
(c) If all groups together must contain exactly [i]seven [/i] distinct numbers, what is the greatest number of groups that you can make?
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 $A$.
Let $n > 1$ be an integer and $\Omega=\{1,2,...,2n-1,2n\}$ the set of all positive integers that are not larger than $2n$.
A nonempty subset $S$ of $\Omega$ is called [i]sum-free[/i] if, for all elements $x, y$ belonging to $S, x + y$ does not belong to $S$. We allow $x = y$ in this condition.
Prove that $\Omega$ has more than $2^n$ distinct [i]sum-free[/i] subsets.
Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection.
a) Show that the intersection of all subsets of $F$ is not empty.
b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?
Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.
Two distinct positive integers are called [i]close [/i] if their greatest common divisor equals their difference. Show that for any $n$, there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.
Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$, $x+|x| \in X$.