Found problems: 110
2023 239 Open Mathematical Olympiad, 7
Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?
2001 Miklós Schweitzer, 1
Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.
2020 Macedonian Nationаl Olympiad, 4
Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met:
(i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ;
(ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$.
Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.
2007 Indonesia TST, 4
Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.
2007 Miklós Schweitzer, 2
We partition the $n$-element subsets of an $n^2+n-1$-element set into two classes. Prove that one of the classes contains $n$-many pairwise disjunct sets.
(translated by Miklós Maróti)
1989 Putnam, B4
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?
2023 239 Open Mathematical Olympiad, 8
Let $n{}$ and $k{}$ be natural numbers, with $n > 2k$. In the deck of cards, each card contains a subset of the set $\{1, 2, \ldots , n\}$ consisting of at least $k+1$, but no more than $n-k$ elements. Each $m$-element set is written exactly on $m-k$ cards. Is it possible to split these cards into $n- 2k$ stacks so that in each stack all subsets on the cards are different, and any two of them intersect?
2023 Hong Kong Team Selection Test, Problem 3
Let $n\ge 4$ be a positive integer. Consider any set $A$ formed by $n$ distinct real numbers such that the following condition holds: for every $a\in A$, there exist distinct elements $x, y, z \in A$ such that $\left| x-a \right|, \left| y-a \right|, \left| z-a \right| \ge 1$. For each $n$, find the greatest real number $M$ such that $\sum_{a\in A}^{}\left| a \right|\ge M$.
2012 IFYM, Sozopol, 5
We denote with $p_n(k)$ the number of permutations of the numbers $1,2,...,n$ that have exactly $k$ fixed points.
a) Prove that $\sum_{k=0}^n kp_n (k)=n!$.
b) If $s$ is an arbitrary natural number, then:
$\sum_{k=0}^n k^s p_n (k)=n!\sum_{i=1}^m R(s,i)$,
where with $R(s,i)$ we denote the number of partitions of the set $\{1,2,...,s\}$ into $i$ non-empty non-intersecting subsets and $m=min(s,n)$.
2010 IMC, 4
Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.
2019 Romania Team Selection Test, 4
Let be two natural numbers $ m,n, $ and $ m $ pairwise disjoint sets of natural numbers $ A_0,A_1,\ldots ,A_{m-1}, $ each having $ n $ elements, such that no element of $ A_{i\pmod m} $ is divisible by an element of $ A_{i+1\pmod m} , $ for any natural number $ i. $
Determine the number of ordered pairs
$$ (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j $$
such that $ a|b $ and such that $ \{ a,b \}\not\in A_k, $ for any $ k\in\{ 0,1,\ldots ,m-1 \} . $
[i]Radu Bumbăcea[/i]
2007 Miklós Schweitzer, 6
For which subsets $A\subset \mathbb R$ is it true that whenever $0\leq x_0 < x_1 < \cdots < x_n\leq 1$, $n=1,2, \ldots$, then there exist $y_j\in A$ numbers, such that $y_{j+1}-y_j>x_{j+1}-x_j$ for all $0\leq j < n$.
(translated by Miklós Maróti)
2008 Miklós Schweitzer, 5
Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which
$$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$
and show that there is no set for which the error term is $o(\log\log x)$ in the above formula.
(translated by Miklós Maróti)
2017 China Team Selection Test, 3
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:
$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;
$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;
$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$
$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.
Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.
2020 Bangladesh Mathematical Olympiad National, Problem 9
Bristy wants to build a special set $A$. She starts with $A=\{0, 42\}$. At any step, she can add an integer $x$ to the set $A$ if it is a root of a polynomial which uses the already existing integers in $A$ as coefficients. She keeps doing this, adding more and more numbers to $A$. After she eventually runs out of numbers to add to $A$, how many numbers will be in $A$?
2007 Indonesia TST, 4
Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.
2018 China Team Selection Test, 4
Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.
2017 Polish MO Finals, 4
Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.
2016 Postal Coaching, 1
The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets: $\mathbb R^+ = A \cup B\cup C$ and $A \cap B = B \cap C = C \cap A = \emptyset$ whereas none of $A, B, C$ is empty.
[list=a][*] Show that one can choose $a \in A, b \in B$ and $c \in C$ such that $a,b, c$ are the sides of a triangle.
[*] Is it always possible to choose three numbers from three different sets $A,B,C$ such that these three numbers are the sides of a right-angled triangle?[/list]
2011 Laurențiu Duican, 2
Consider a finite set $ A, $ and two functions $ f,g: A\longrightarrow A. $ Prove that:
$$ |\{ x\in A| g(f(x))\neq x \} | =|\{ x\in A| f(g(x))\neq x \} | $$
[i]Cristinel Mortici[/i]
2021 Iran Team Selection Test, 6
Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where :
$$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$
Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]
2014 Greece JBMO TST, 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) $n=2018$
2000 Tuymaada Olympiad, 1
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?
2022 Vietnam TST, 6
Given a set $A=\{1;2;...;4044\}$. They color $2022$ numbers of them by white and the rest of them by black. With each $i\in A$, called the [b][i]important number[/i][/b] of $i$ be the number of all white numbers smaller than $i$ and black numbers larger than $i$. With every natural number $m$, find all positive integers $k$ that exist a way to color the numbers that can get $k$ important numbers equal to $m$.
2017 China Team Selection Test, 3
Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions:
($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$;
($ii$)$2017|x_1+...+x_{100}$;
($iii$)$2017|x_1^2+...+x_{100}^2$.