Problem 4. We call a set interesting if it consists of 2022 strictly positive real numbers with the property that, when we write its elements in ascending order, no element is equal to the arithmetic mean of its neighbors.

For any set $A$, we attach the set $\widetilde{A}=\{x+y \mid x, y \in A\}$.

a) Determine the maximum possible cardinality of a set $\widetilde{A}$ when $A$ ranges over all interesting sets.

b) Determine the minimum possible cardinality of a set $\widetilde{A}$ when $A$ ranges over all interesting sets.
Solution. a) The maximum cardinality of a set $\widetilde{A}$ is obtained when all sums $x+y$, where $x, y \in A$, are distinct. In this case, $|\widetilde{A}|=\frac{2022 \cdot 2023}{2}$. $1 p$

This maximum is achieved, for example, for the interesting set

$$
A=\left\{1,3,3^{2}, \ldots, 3^{2021}\right\}
$$

$1 p$

b) Let $a_{1}<a_{2}<\ldots<a_{2022}$ be the elements of an interesting set $A$.

Then the sums $a_{1}+a_{2}, a_{2}+a_{2}$, and $a_{1}+a_{3}$ are distinct; we denote by $A_{1}$ the set formed by these three numbers. Similarly, the sums $a_{2}+a_{3}, a_{3}+a_{3}$, and $a_{2}+a_{4}$ are distinct; we denote by $A_{2}$ the set formed by these numbers. We observe that $A_{1}$ and $A_{2}$ are disjoint, since $a_{2}+a_{3}$, the smallest element of $A_{2}$, is greater than any element of $A_{1}$.

In the same manner, we define the sets $A_{3}, A_{4}, \ldots, A_{2020}$, which are pairwise disjoint. Then

$$
\left\{a_{1}+a_{1}\right\} \cup A_{1} \cup A_{2} \cup \ldots \cup A_{2020} \cup\left\{a_{2021}+a_{2022}, a_{2022}+a_{2022}\right\} \subset \widetilde{A}
$$

It follows that $|\widetilde{A}| \geqslant 1+3 \cdot 2020+2=6063$.

$3 p$

This minimum is achieved, for example, for the interesting set

$$
A=\{1,2,4,5,7,8, \ldots, 3031,3032\}
$$

The attached set is $\widetilde{A}=\{2,3,4,5, \ldots, 6064\}$, of cardinality 6063. $2 p$