1. A set of integers $T$ is orensan if there exist integers $a<b<c$ such that $a$ and $c$ belong to $T$ and $b$ does not belong to $T$. Find the number of subsets $T$ of $\{1,2, \ldots, 2019\}$ that are orensan.
Solution. The number of subsets of $\{1,2, \cdots, 2019\}$ is $2^{2019}$ as is well known. Let's now count the number of these sets that do NOT have the desired property. Clearly, the empty set, and the subsets of $\{1,2, \ldots, 2019\}$ with a single element do not have the desired property, since it is impossible to choose distinct $a, c$ in them. There are in total

$$
\left(\begin{array}{c}
2019 \\
0
\end{array}\right)+\left(\begin{array}{c}
2019 \\
1
\end{array}\right)=1+2019=2020
$$

of these sets. Any other subset that does not have the desired property must be formed by consecutive elements of $\{1,2, \ldots, 2019\}$. Indeed, if $m, M$ are the minimum and maximum of $T$ respectively, for $T$ to NOT have the stated property, all elements $m, m+1, m+2, \ldots, M$ must be in $T$, because if any $k$ were not, we could take $a=m$, $b=k$ and $c=M$, satisfying the property. The number of such subsets is $\left(\begin{array}{c}2019 \\ 2\end{array}\right)$. Indeed, given such a subset, its minimum $m$ and its maximum $M$ uniquely determine it, and there are $\left(\begin{array}{c}2019 \\ 2\end{array}\right)$ different ways to choose two numbers from the set $\{1,2, \ldots, 2019\}$, such that the smaller one will be $m$, and the larger one $M$.

Finally, the total number of subsets $T$ of $\{1,2, \cdots, 2019\}$ with the desired property is

$$
2^{2019}-2020-\left(\begin{array}{c}
2019 \\
2
\end{array}\right)=2^{2019}-2039191
$$