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National and Regional Contests
China Contests
China Team Selection Test
2023 China Team Selection Test
P1
P1
Part of
2023 China Team Selection Test
Problems
(1)
A Special Cyclic Polygon
Source: 2023 China Team Selection Test Day 1 Problem 1
3/14/2023
Given an integer
n
⩾
2
n \geqslant 2
n
⩾
2
. Suppose there is a point
P
P
P
inside a convex cyclic
2
n
2n
2
n
-gon
A
1
…
A
2
n
A_1 \ldots A_{2n}
A
1
…
A
2
n
satisfying
∠
P
A
1
A
2
=
∠
P
A
2
A
3
=
…
=
∠
P
A
2
n
A
1
,
\angle PA_1A_2 = \angle PA_2A_3 = \ldots = \angle PA_{2n}A_1,
∠
P
A
1
A
2
=
∠
P
A
2
A
3
=
…
=
∠
P
A
2
n
A
1
,
prove that
∏
i
=
1
n
∣
A
2
i
−
1
A
2
i
∣
=
∏
i
=
1
n
∣
A
2
i
A
2
i
+
1
∣
,
\prod_{i=1}^{n} \left|A_{2i - 1}A_{2i} \right| = \prod_{i=1}^{n} \left|A_{2i}A_{2i+1} \right|,
i
=
1
∏
n
∣
A
2
i
−
1
A
2
i
∣
=
i
=
1
∏
n
∣
A
2
i
A
2
i
+
1
∣
,
where
A
2
n
+
1
=
A
1
A_{2n + 1} = A_1
A
2
n
+
1
=
A
1
.
geometry
China TST