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Kvant 2021
M2680
M2680
Part of
Kvant 2021
Problems
(1)
Equality in regular (2^n-1)-gon
Source: Kvant Magazine No. 11-12 2021 M2680
3/9/2023
Let
n
>
1
n>1
n
>
1
be a natural number and
A
0
A
1
…
A
2
n
−
2
A_0A_1\ldots A_{2^n-2}
A
0
A
1
…
A
2
n
−
2
be a regular polygon. Prove that
1
A
0
A
1
=
1
A
0
A
2
+
1
A
0
A
4
+
1
A
0
A
8
+
⋯
+
1
A
0
A
2
n
−
1
.
\frac{1}{A_0A_1}=\frac{1}{A_0A_2}+\frac{1}{A_0A_4}+\frac{1}{A_0A_8}+\cdots+\frac{1}{A_0A_{2^{n-1}}}.
A
0
A
1
1
=
A
0
A
2
1
+
A
0
A
4
1
+
A
0
A
8
1
+
⋯
+
A
0
A
2
n
−
1
1
.
Proposed by Le Hoang and Ngoc Thai (Vietnam)
geometry
Kvant