Found problems: 233
2017 China Team Selection Test, 3
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
2011 Ukraine Team Selection Test, 4
Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount.
For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.
1968 Putnam, A3
Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that
(1) $A_1 =\emptyset$.
(2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$
2018 Junior Regional Olympiad - FBH, 3
In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of them also goes to spanish language. $10$ students who study spanish also study german. $12$ students who study german also do sports. Only $4$ students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?
2015 Indonesia MO Shortlist, N6
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.
KoMaL A Problems 2021/2022, A. 811
Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\ldots,m$ satisfying the following conditions:
[list=1]
[*]$B_i\subset A,$ $|B_i|=k,$
[*]$C_i\subset B_i$ (there is no additional condition for the number of elements in $C_i$), and
[*]$B_i\cap C_j\neq B_j\cap C_i$ for all $i\neq j.$
[/list]
2018 Malaysia National Olympiad, B2
A subset of $\{1, 2, 3, ... ... , 2015\}$ is called good if the following condition is fulfilled: for any element $x$ of the subset, the sum of all the other elements in the subset has the same last digit as $x$.
For example, $\{10, 20, 30\}$ is a good subset since $10$ has the same last digit as $20 + 30 = 50$, $20$ has the same last digit as $10 + 30 = 40$, and $30$ has the same last digit as $10 + 20 = 30$.
(a) Find an example of a good subset with 400 elements.
(b) Prove that there is no good subset with 405 elements.
2014 India PRMO, 8
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have?
2020-2021 Winter SDPC, #7
Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.
2021 Science ON all problems, 3
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]
2023 AMC 12/AHSME, 24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
2005 National Olympiad First Round, 24
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community?
$
\textbf{(A)}\ 120
\qquad\textbf{(B)}\ 380
\qquad\textbf{(C)}\ 570
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$
2011 Silk Road, 1
Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$, where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions:
$(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$, i.e. any two distinct sets contain exactly one element in common;
$(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$, i.e. any four different sets contain no common element.
Where $| S |$ means the number of elements of $S$.
2011 VTRMC, Problem 6
Let $S$ be a set with an asymmetric relation $<$; this means that if $a,b\in S$ and $a<b$, then we do not have $b<a$. Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a,b\in S$, then $a<b$ if and only if $a\prec b$, and if $x,y\in T$ with $x\prec y$, then there exists $t\in T$ such that $x\prec t\prec y$.
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2000 Mexico National Olympiad, 3
Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way:
Write down some elements of $A$ one after another without repeating, write a sign $+ $ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$.
For example, if $A = \{2,8,13,20\}$, numbers $8$ and $14 = 20-2+8$ are elements of $A'$.
Set $A''$ is constructed from $A'$ in the same manner.
Find the smallest possible number of elements of $A$, if $A''$ contains all the integers from $1$ to $40$.
2017 Bosnia And Herzegovina - Regional Olympiad, 3
Let $S$ be a set of $6$ positive real numbers such that
$\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S$ or $a-b \in S$
Prove that if we sort these numbers in ascending order, then they form an arithmetic progression
2021 Balkan MO Shortlist, C1
Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple.
Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.
2004 Korea Junior Math Olympiad, 2
For $n\geq3$ define $S_n=\{1, 2, ..., n\}$. $A_1, A_{2}, ..., A_{n}$ are given subsets of $S_n$, each having an even number of elements. Prove that there exists a set $\{i_1, i_2, ..., i_t\}$, a nonempty subset of $S_n$ such that
$$A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset$$
(For two sets $A, B$, we define $\Delta$ as $A \Delta B=(A\cup B)-(A\cap B)$)
2018 JBMO Shortlist, C1
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 $\{1,2,... ,n\}$.
2009 Ukraine Team Selection Test, 6
Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.
2016 Indonesia TST, 1
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 $n$.
2018 Brazil Team Selection Test, 1
Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\
3 \mid f(S)}}} f(S)$$
where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.
2022 IMC, 4
Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of
$\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever
$1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.
2022 Turkey EGMO TST, 2
We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.