Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than: a) $11$. b) $13$.
Problem
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Tags: combinatorics, algebra
BackToSchool
25.10.2022 12:31
There are total $8$ sums and each integer shall be used for three times.
$$ \frac {(1+8)\cdot 8}{2} \cdot 3 \ge 12 \cdot 8 \Leftrightarrow 108 > 96$$
$$ \frac {(1+8)\cdot 8}{2} \cdot 3 \ge 14 \cdot 8 \Rightarrow 108 > 112$$
Pi-rate_91
25.10.2022 19:19
parmenides51 wrote: Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than: a) $11$. b) $13$. a) yes b) no