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Suppose

n>2

and let

A_1,\dots,A_n

be points on the plane such that no three are collinear. (a) Suppose

M_1,\dots,M_n

be points on segments

A_1A_2,A_2A_3,\dots ,A_nA_1

respectively. Prove that if

B_1,\dots,B_n

are points in triangles

M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n

respectively then

|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|

Where

|XY|

means the length of line segment between

X

and

Y

.
(b) If

X

Y

and

Z

are three points on the plane then by

H_{XYZ}

we mean the half-plane that it's boundary is the exterior angle bisector of angle

\hat{XYZ}

and doesn't contain

X

and

Z

,having

Y

crossed out. Prove that if

C_1,\dots ,C_n

are points in

{H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}

then

|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|

Time allowed for this problem was 2 hours.

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