Found problems: 149
2018 Regional Competition For Advanced Students, 3
Let $n \ge 3$ be a natural number.
Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two.
[i]Proposed by Walther Janous[/i]
2014 Contests, 4
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
1998 Singapore Team Selection Test, 2
Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -
2009 Ukraine Team Selection Test, 3
Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection.
a) Show that the intersection of all subsets of $F$ is not empty.
b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?
2014 Czech-Polish-Slovak Match, 6
Let $n \ge 6$ be an integer and $F$ be the system of the $3$-element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition:
for every $1 \le i < j \le n$ there is at least $ \lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in F$ such that $i, j \in A$.
Prove that for some integer $m \ge 1$ exist the mutually disjoint subsets $A_1, A_2 , ... , A_m \in F $ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5 $
(Poland)
PS. just in case my translation does not make sense,
I leave the original in Slovak, in case someone understands something else
2021 Indonesia TST, C
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
1975 Czech and Slovak Olympiad III A, 6
Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties:
1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$
2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\]
Show that in fact \[\mathbf M=\mathbb R^2.\]
2020 Malaysia IMONST 1, 19
A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?
2012 Switzerland - Final Round, 5
Let n be a natural number. Let $A_1, A_2, . . . , A_k$ be distinct $3$-element subsets of $\{1, 2, . . . , n\}$ such that $|A_i \cap A_j | \ne 1$ for all $1 \le i, j \le k$. Determine all $n$ for which there are $n$ such that these subsets exist.
[hide=original wording of last sentence]Bestimme alle n, fur die es n solche Teilmengen gibt.[/hide]
2009 Postal Coaching, 1
Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property:
if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.
2001 Abels Math Contest (Norwegian MO), 2
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?
1977 Chisinau City MO, 136
We represent the number line $R$ as the union of two non-empty sets $A, B$ different from $R$. Prove that one of the sets $A, B$ does not have the following property: the difference of any elements of the set belongs to the same set.
1956 Putnam, B2
Suppose that each set $X$ of points in the plane has an associated set $\overline{X}$ of points called its cover. Suppose further that (1) $\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y$ for all sets $X,Y$ . Show that i) $\overline{X} \supset X$, ii) $\overline{\overline{X}}=\overline{X}$ and iii) $X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.$ Prove also that these three statements imply (1).
2016 Latvia Baltic Way TST, 9
The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third.
[hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo.
[/hide]
2009 Thailand Mathematical Olympiad, 2
Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.
2008 Thailand Mathematical Olympiad, 9
Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$
1996 IMO Shortlist, 3
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
1988 Tournament Of Towns, (177) 3
The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements?
(S. Fomin , Leningrad)
2012 Danube Mathematical Competition, 4
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.
2014 India PRMO, 20
What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
1989 Tournament Of Towns, (238) 2
Consider all the possible subsets of the set $\{1,2,..., N\}$ which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is $(N + 1)! - 1$.
(Based on idea of R.P. Stanley)
2010 Benelux, 1
A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets.
(A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.)
[i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]
1969 Vietnam National Olympiad, 1
A graph $G$ has $n + k$ vertices. Let $A$ be a subset of $n$ vertices of the graph $G$, and $B$ be a subset of other $k$ vertices. Each vertex of $A$ is joined to at least $k - p$ vertices of $B$. Prove that if $np < k$ then there is a vertex in $B$ that can be joined to all vertices of $A$.
2012 Belarus Team Selection Test, 3
Define $M_n = \{1,2,...,n\}$, for any $n\in N$. A collection of $3$-element subsets of $M_n$ is said to be [i]fine [/i] if for any coloring of elements of $M_n$ in two colors there is a subset of the collection all three elements of which are of the same color.
For any $n\ge 5$ find the minimal possible number of the $3$-element subsets of $M_n$ in the fine collection.
(E. Barabanov, S. Mazanik, I. Voronovich)