In a certain convex $n$-gon ( $n>3$ ) all distances between vertices are distinct.

a) We will call a vertex uninteresting if the closest vertex to it is adjacent to it. What is the smallest possible number of uninteresting vertices (for a given $n$)?

b) We will call a vertex unusual if the farthest vertex from it is adjacent to it. What is the largest possible number of unusual vertices (for a given $n$)?

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a) Example. Let's take a segment $AB$ and a convex broken line $l$ close to it with the same ends and with links of the same length. The broken line $l$ and its symmetric counterpart relative to $AB$ form a convex polygon, in which the only uninteresting vertices will be $A$ and $B$. Thus, we can obtain a polygon with any even number of vertices. Now, replace $l$ (only on one side of $AB$) with a similar broken line that has one more link. This results in a polygon with any odd number of vertices greater than 3, in which the uninteresting vertices will also be only $A$ and $B$. In both cases, a small movement of the vertices can ensure that all distances between them are different.

Estimate. Let $A$ be an interesting vertex of a convex $n$-gon, $B$ be the nearest vertex to it. The segment $AB$ cuts the polygon into a "right" and a "left" part. In the right part, there is some vertex $C$ different from $A$ and $B$.

Suppose it is interesting and $D$ is the nearest vertex to it. If $D$ is located in the left part of the polygon, then in the convex quadrilateral $ABCD$

$AB + CD < BC + AD$, and we slightly "break" the side $AC$ so that a convex $n$-gon is obtained. Then the unusual vertices are $A, B, C$ and only they.

Estimate. Let $X$ be an unusual vertex, $Y$ be the farthest vertex from it, $Z$ be the other adjacent vertex to $Y$.

Then $XZ < AE, DE > BD$, that is, the sum of the diagonals is less than the sum of the opposite sides, which is impossible.

In the opposite case, a similar contradiction arises in the quadrilateral $BCDE$.

Cases of $E$ coinciding with $A$ or with $B$ lead to a contradiction similarly.

## Answer

a) 2; b) 3 vertices.