Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$. Show that there are two distinct elements of $A$, having the same sum of their elements.

Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs. $(1)$ Calculate $a(m,n)$ and $b(m,n).$ $(2)$ Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$

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?

The European zoos with exactly $100$ types of species each are separated into two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A, B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. Prove that we can colour the cages in $3$ colours (all animals of the same type live in the same cage) such that no zoo has cages of only one colour

Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions. (i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$ (ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$ (iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$ (iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$

Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?

Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8} (1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$ Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S? (Notice $S$ and S are different.) (2)In $S$, how many permutations are there which satisfies "For all $k=1,2,...,7$,the digit after k is [b]not[/b] (k+1)"?

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 $a_1+\cdots+a_r=0$.

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.

Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \in A$ , then $n \le 2018$. $b.)$ If $S \subset A$ such that $|S|=3$, then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$ What is the maximum cardinality of $A$ ?

Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.

Finite set $\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}$ is given. If $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$, show that $$\sum_{i=1}^n |a_i-b_i|=n^2$$

Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold: - If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$ - If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$. What is the maximum possible number of elements in $S$? $ \textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675 \qquad $

Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.

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 $A(n) = A(n + 1)$

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 $(0, 1)$.

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]

A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?

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 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 $n=2k$ [/list]

Let $S_{1},$ $S_{2},$ $...,$ $S_{n}$ be finitely many subsets of $\mathbb{N}$ such that $S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}.$ Prove that there exists some $k\in\left\{ 1,2,...,n\right\} $ such that for each positive integer $m,$ the set $S_{k}$ contains infinitely many multiples of $m.$

Let $N$ be the set of positive integers. If $A,B,C \ne \emptyset$, $A \cap B = B \cap C = C \cap A = \emptyset$ and $A \cup B \cup C = N$, we say that $A,B,C$ are partitions of $N$. Prove that there are no partitions of $N, A,B,C$, that satisfy the following: (i) $\forall a \in A, b \in B$, we have $a + b + 1 \in C$ (ii) $\forall b \in B, c \in C$, we have $b + c + 1 \in A$ (iii) $\forall c \in C, a \in A$, we have $c + a + 1 \in B$

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 $x$).

$(a)$ 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 $[0,1).$

Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.