Found problems: 110
2019 Taiwan APMO Preliminary Test, P5
Find the minimum positive integer $n$ such that for any set $A$ with $n$ positive intergers has $15$ elements which sum is divisible by $15$.
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 $\sum_{a\in A}^{}\left| a \right|\ge M$.
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?
2012 IFYM, Sozopol, 1
Let $A_n$ be the set of all sequences with length $n$ and members of the set $\{1,2…q\}$. We denote with $B_n$ a subset of $A_n$ with a minimal number of elements with the following property: For each sequence $a_1,a_2,...,a_n$ from $A_n$ there exist a sequence $b_1,b_2,...,b_n$ from $B_n$ such that $a_i\neq b_i$ for each $i=1,2,....,n$. Prove that, if $q>n$, then $|B_n |=n+1$.
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)
2016 VJIMC, 2
Let $X$ be a set and let $\mathcal{P}(X)$ be the set of all subsets of $X$. Let $\mu: \mathcal{P}(X) \to \mathcal{P}(X)$ be a map with the property that $\mu(A \cup B) = \mu(A) \cup \mu(B)$ whenever $A$ and $B$ are disjoint subsets of $X$. Prove that there exists $F \subset X$ such that $\mu(F) = F$.
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 $\{B_j\}_{j=1}^n$.
2019 Jozsef Wildt International Math Competition, W. 42
For $p$, $q$, $l$ strictly positive real numbers, consider the following problem: for $y \geq 0$ fixed, determine the values $x \geq 0$ such that $x^p - lx^q \leq y$. Denote by $S(y)$ the set of solutions of this problem. Prove that if one has $p < q$, $\epsilon \in (0, l^\frac{1}{p-q})$, $0 \leq x \leq \epsilon$ and $x \in S(y)$, then $$x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}$$
2000 Iran MO (2nd round), 1
Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.
1996 Tuymaada Olympiad, 2
Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?
2016 IMC, 4
Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:
(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;
(ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.
Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
1975 Miklós Schweitzer, 1
Show that there exists a tournament $ (T,\rightarrow)$ of cardinality $ \aleph_1$ containing no transitive subtournament of size $ \aleph_1$. ( A structure $ (T,\rightarrow)$ is a $ \textit{tournament}$ if $ \rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $ (T,\rightarrow)$ is transitive if $ \rightarrow$ is transitive, that is, if it orders $ T$.)
[i]A. Hajnal[/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]
KoMaL A Problems 2020/2021, A. 797
We call a system of non-empty sets $H$ [i]entwined[/i], if for every disjoint pair of sets $A$ and $B$ in $H$ there exists $b\in B$ such that $A\cup\{b\}$ is in $H$ or there exists $a\in A$ such that $B\cup\{a\}$ is in $H.$
Let $H$ be an entwined system of sets containing all of $\{1\},\{2\},\ldots,\{n\}.$ Prove that if $n>k(k+1)/2,$ then $H$ contains a set with at least $k+1$ elements, and this is sharp for every $k,$ i.e. if $n=k(k+1),$ it is possible that every set in $H$ has at most $k$ elements.
1990 China Team Selection Test, 4
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
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 $m=min(s,n)$.
Russian TST 2018, P2
Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.
2015 Postal Coaching, Problem 3
Let $A$ be a non empty subset of positive reals such that for every $a,b,c \in A$, the number $ab+bc+ca$ is rational.
Prove that $\frac{a}{b}$ is a rational for every $a,b \in A$.
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]
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?
2022 IFYM, Sozopol, 4
Let $x_1,\dots ,x_n$ be real numbers. We look at all the $2^{n-1}$ possible sums between some of the numbers. If the number of different sums is at least $1.8^n$, prove that the number of sums equal to $2022$ is no more than $1.67^n$.
2007 USAMO, 3
Let $S$ be a set containing $n^{2}+n-1$ elements, for some positive integer $n$. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.
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 $n$.
2010 ISI B.Stat Entrance Exam, 3
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.
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)