Found problems: 295
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 $S$.
For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors.
Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.
Let $A=\{1,2,\cdots ,100\}$. Let $S$ be a subset of power set of $A$ such that any two elements of $S$ has nonzero intersection (Note that elements of $S$ are actually some subsets of $A$). Then the maximum possible cardinality of $S$ is
[list=1]
[*] $2^{99}$
[*] $2^{99}+1$
[*] $2^{99}+2^{98}$
[*] None of these
[/list]
Suppose that $S$ is a subset of $\{1, 2, 3,...,25\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain?
$\textbf{(A) }12 \qquad \textbf{(B) }13 \qquad \textbf{(C) }14 \qquad \textbf{(D) }15 \qquad \textbf{(E) }16$
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 $\max(\deg f,c)$.
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]
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 $k\ge 1$ be an integer and $\mathsf S$ be a family of 2-element subsets of the index set $\{1,\ldots,2k\}$ with the following property: if $\mathsf M_1,\ldots,\mathsf M_{2k}$ are arbitrary sets such that \[\mathsf M_i\cap\mathsf M_j\neq\emptyset\quad\Leftrightarrow\quad\{i,j\}\in\mathsf S,\] then the union $\mathsf M_1\cup\ldots\cup\mathsf M_{2k}$ contains at least $k^2$ elements. Show that there is a suitable family $\mathsf S$ for any integer $k\ge1.$
Let $A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}$, $A + A = \{a + b |a,b \in A\}$,$A \cdot A =\{a \cdot b | a, b \in A\}$.
Prove that:
i) $A + A \ne A \cdot A$
ii) $(A + A) \cap N = (A \cdot A) \cap N$.
Vasile Pop
Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$.
Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .
Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.
Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .
Five people form several commissions to prepare a competition. Here any commission must be nonempty and any two commissions cannot contain the same members. Moreover, any two commissions have at least one common member.
There are already $14$ commissions. Prove that at least one additional commission can be formed.
There are $12$ members in a club. The members created some small groups, which satisfy the following:
- The small group consists of $3$ or $4$ people.
- Also, for two arbitrary members, there exists exactly one small group that has both members.
Prove that all members are in the same number of small groups.
A set $S$ is called [i]neighbouring [/i] if it has the following two properties:
a) $S$ has exactly four elements
b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$.
Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.
A nonempty set $S$ is called [i]Bally[/i] if for every $m\in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of Bally subsets of $\{1, 2, . . . , 2020\}$.
Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have
(i) $168 \in I_i$;
(ii) $\forall x, y \in I_i$, we have $x-y \in I_i$.
Determine the number of $k-chain$ in total.
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that
$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$
$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$
where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
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 $2422$?
Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.
Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements
It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that:
$a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled
$b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled
Which numbers must be expelled such that sum of numbers remaining in set stays minimal
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?