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.

Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned in to $3$ subsets with equal sums. Find the number of elements of $E$.

An infinite fairytale is a book with pages numbered $1,2,3,\ldots$ where all natural numbers appear. An author wants to write an infinite fairytale such that a new dwarf is introduced on each page. Afterward, the page contains several discussions between groups of at least two of the already introduced dwarfs. The publisher wants to make the book more exciting and thus requests the following condition: Every infinite set of dwarfs contains a group of at least two dwarfs, who formed a discussion group at some point as well as a group of the same size for which this is not true. Can the author fulfill this condition?

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 $A_S$

Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that (1) $A_1 =\emptyset$. (2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$

Given $I_0 = \{-1,1\}$, define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$, where $y$ ranges over all elements of $I_{n-1}$. Determine the union of the sets $I_n$ over all nonnegative integers $n$.

Let $n$ be a positive integer and $A$ be a family of subsets of the set $\{1,2,...,n\},$ none of which contains another subset from A . Find the largest possible cardinality of $A$ .

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 $a,b\in\mathbb{R}$.

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]

The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?

A subset of the real numbers has the property that for any two distinct elements of it such as $x$ and $y$, we have $(x+y-1)^2 = xy+1$. What is the maximum number of elements in this set? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$

$(a)$ Is it possible to partition the segment $[0,1]$ into two sets $A$ and $B$ and to define a continuous function $f$ such that for every $x\in A \ f(x)$ is in $B$, and for every $x\in B \ f(x)$ is in $A$? $(b)$ The same question with $[0,1]$ replaced by $[0,1).$

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

Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$

A collection $F$ of distinct (not necessarily non-empty) subsets of $X = \{1,2,\ldots,300\}$ is [i]lovely[/i] if for any three (not necessarily distinct) sets $A$, $B$ and $C$ in $F$ at most three out of the following eight sets are non-empty \begin{align*}A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overline{B} \cap C, \ \ \ A \cap B \cap \overline{C}, \\ \overline{A} \cap \overline{B} \cap C, \ \ \ \overline{A} \cap B \cap \overline {C}, \ \ \ A \cap \overline{B} \cap \overline{C}, \ \ \ \overline{A} \cap \overline{B} \cap \overline{C} \end{align*} where $\overline{S}$ denotes the set of all elements of $X$ which are not in $S$. What is the greatest possible number of sets in a lovely collection?

Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple. Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.

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 $\max(\deg f,c)$.

We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?

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]

A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?