67th NMO Selection Tests for BMO and IMO

Recall that a necessary and sufficient condition for $k\ge 3$ positive real numbers $s_1,\dots ,s_k$ to be the side lengths of a non-degenerate planar $k$-gon is that a maximal $s_i$ be less than the sum of the other $s_j$.

a) The answer is in the affirmative. Given pairwise distinct positive real numbers ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ less than $1/2$, we show that the sets $S=\{1,2,4,\dots ,2^{n-5},2^{n-4}+{\epsilon }_1,2^{n-4}+{\epsilon }_2,2^{n-4}+{\epsilon }_3,2^{n-3}-1/2\}$ and $S\setminus \{2^{n-3}-1/2\}$ are both multipolygonal.

Split any of the two sets into two subsets each of which has at least three elements, let $A$ be the part containing at least two of the $2^{n-4}+{\epsilon }_i$, and let $B$ be the other part. The set $A$ is polygonal since its maximal element is either one of the $2^{n-4}+{\epsilon }_i$ or $2^{n-3}-1/2$, each of which is smaller than the sum of other elements in $A$. To prove that $B$ is not polygonal, notice that its maximal element is either $2^{n-3}-1/2$, or one of the $2^{n-4}+{\epsilon }_i$, or some $2^k$, $k\le n-5$. In the first case, the sum of all other elements in $B$ is less than $1+2+\cdots +2^{n-5}+2^{n-4}+{\epsilon }_i=2^{n-3}-1+{\epsilon }_i<2^{n-3}-1/2$; in the second case, this sum does not exceed $1+2+\cdots +2^{n-5}=2^{n-4}-1<2^{n-4}+{\epsilon }_i$; and in the third case, this sum is at most $1+2+\cdots +2^{k-1}=2^k-1<2^k$. Consequently, $B$ is not polygonal.