Found problems: 295
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 $A.$
Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.
The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ :
$[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$.
Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?
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 $n$.
$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$.
[b]Condition[/b]
(1) $1 \in A$
(2) If $x \in A$, then $x^2 \in A$.
(3) If $(x-3)^2 \in A$, then $x \in A$.
The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$.
Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$.
[i](I. Voronovich)[/i]
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$,
$(\text i)$ $x*z=y*z$ implies $x=y$
$(\text{ii})$ $x*(y*z)\ne(x*y)*z$
$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition.
[b]Condition[/b]
The sum of the $1998$ elements is again an element of $T$.
Let $p$ be a nonnegative integer and let $n=2^p$. Consider all subsets $A$ of the set $\{1,2,\ldots,n\}$ with the property that, whenever $x\in A$, $2x\notin A$. Find the maximum number of elements that such a set $A$ can have.
On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$.
Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$
Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.
Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.
Find all real numbers $m$ such that $$\frac{1-m}{2m} \in \{x\ |\ m^2x^4+3mx^3+2x^2+x=1\ \forall \ x\in \mathbb{R} \}$$
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.
$a)$ Prove that for all positive integers $n \geq 3$ holds:
$$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$ where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent
$b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers
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 $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.
Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set
$S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.
$a)$ Prove that for all positive integers $n$ exists a set $M_n$ of positive integers with exactly $n$ elements and:
$i)$ Arithmetic mean of arbitrary non-empty subset of $M_n$ is integer
$ii)$ Geometric mean of arbitrary non-empty subset of $M_n$ is integer
$iii)$ Both arithmetic mean and geometry mean of arbitrary non-empty subset of $M_n$ is integer
$b)$ Does there exist infinite set $M$ of positive integers such that arithmetic mean of arbitrary non-empty subset of $M$ is integer
Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$
If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$
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
Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that:
[LIST]
[*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*]
[*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*]
[*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.
[/LIST]
Let $A$ be a set of positive integers having the following property:
for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.
Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.
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 $E$.