Olympiad problems

2016 USAMO, 1

Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.

2016 USAJMO, 3

Tags: USAMO , Problem Sets , 2016 USAMO

Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.

2017 IMAR Test, 3

Tags: Problem Sets , combinatorics

We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.

2019 Spain Mathematical Olympiad, 1

Tags: Spain , Problem Sets , combinatorics

An integer set [i][b]T[/b][/i] is orensan if there exist integers[b] a<b<c[/b], where [b]a [/b]and [b]c[/b] are part of [i][b]T[/b][/i], but [b]b[/b] is not part of [b][i]T[/i][/b]. Count the number of subsets [b][i]T[/i][/b] of {1,2,...,2019} which are orensan.