compare sum of sides, regular, 5-gon, 20-gon (2010 All-Siberian Qualifying 11.2)

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compare sum of sides, regular, 5-gon, 20-gon (2010 All-Siberian Qualifying 11.2)

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#1 h May 12, 2021, 7:31 PM

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Regular pentagon and regular $20$ -gon are inscribed in the same circle. Which is greater, the sum of the squares of the lengths of all sides of the pentagon, or the sum of the squares of the lengths of all sides of the $20$ -gon?

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#2 h May 14, 2021, 6:30 PM

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Side of a regular pentagon, radius $R=1$ , is $a_{5}=\frac{1}{2}\sqrt{10-2\sqrt{5}}$ .
Sum of the squares $5 \cdot \frac{10-2\sqrt{5}}{4}$ .

Side of a regular 10-gon, $a_{2n}=\sqrt{2R^{2}-R\sqrt{4R^{2}-(a_{n})^{2}}}$ ,
$a_{10}=\frac{\sqrt{5}-1}{2}$ .

Side of a regular 20-gon, $a_{20}=\sqrt{2-\frac{1}{2}\sqrt{10+2\sqrt{5}}}$ .
Sum of the squares $20 \cdot (2-\frac{1}{2}\sqrt{10+2\sqrt{5}})$ .

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