Found problems: 295
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 $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of $n$.
Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.
Consider the set of natural numbers $ U = \{1,2,3, ..., 6024 \} $ Prove that for any partition of the $ U $ in three subsets with $ 2008 $ elements each, we can choose a number in each subset so that one of the numbers is the sum of the other two numbers.
Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?
Proposed by Shantanu Nene
Define the distance between real numbers $x$ and $y$ by $d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}$ .
Determine (as a union of intervals) the set of real numbers whose distance from $3/2$ is less than $202/100$ .
It is known that for irrational numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and for any positive integer $n$ the following is true:
$$[n\alpha]+[n\beta]=[n\gamma]+[n\delta]$$
Does this mean that sets $\{\alpha,\beta\}$ and $\{\gamma,\delta\}$ are equal? (As usual $[x]$ means the greatest integer not greater than $x$).
Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$
[list=1]
[*] Find $|F|$
[*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$
[*] Find $n$, if $|F|=540$ and $n=2k$
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Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$.
[i] (From "Radu Păun" contest, Radu Miculescu)[/i]
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
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.
The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$. The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$. Prove that there is a natural number, that doesn't belong to $T_{1987}$.
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.
A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements of $A$. $A-B$ is the set of elements that are in $A$ but not $B$. $\text{min}(x,y)$ is the number that is not larger than the other.)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots,n\}$
(ii): For all $1 \le i < j \le m$, $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$
Given a set $S$ of $2024$ natural numbers satisfying the following condition: if you select any $10$ (different) numbers from $S$, then you can select another number from $S$ so that the sum of all $11$ selected numbers is divisible by $10$. Prove that one of the numbers can be thrown out of $S$ so that the resulting set $S'$ of $2023$ numbers satisfies the condition: if you choose any $9$ (different) numbers from $S'$, then you can choose another number from $S'$ so that the sum of all $10$ selected numbers is divisible by $10$.
[i]K.A.Sukhov[/i]
$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$
[list=1]
[*] 360
[*] 364
[*] 365
[*] 366
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Let $S$ be a set of rational numbers with the following properties:
(a) $\frac12$ is an element in $S$,
(b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$.
Prove that $S$ contains all rational numbers in the interval $(0, 1)$.
Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$. (For example, $A = \{1,2,3\}$, then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$.)
A subset $F$ of P $(A)$ is called [i]strong [/i] if the following is true:
If $B_1$ and $B_2$ are elements of $F$, then $B_1 \cup B_2$ is also an element of $F$.
Suppose that $F$ and $G$ are strong subsets of $P (A)$.
a) Is the union $F \cup G$ necessarily strong?
b) Is the intersection $F \cap G$ necessarily strong?
We have many $\text{three-element}$ subsets of a $1000\text{-element}$ set. We know that the union of every $5$ of them has at least $12$ elements. Find the most possible value for the number of these subsets.
Let $M$ be the set of positive odd integers.
For every positive integer $n$, denote $A(n)$ the number of the subsets of $M$ whose sum of elements equals $n$.
For instance, $A(9) = 2$, because there are exactly two subsets of $M$ with the sum of their elements equal to $9$: $\{9\}$ and $\{1, 3, 5\}$.
a) Prove that $A(n) \le A(n + 1)$ for every integer $n \ge 2$.
b) Find all the integers $n \ge 2$ such that $A(n) = A(n + 1)$