MathDB

Let

m

and

n

be positive integers. A sequence of points

(A_0,A_1,\ldots,A_n)

on the Cartesian plane is called interesting if

A_i

are all lattice points, the slopes of

OA_0,OA_1,\cdots,OA_n

are strictly increasing (

O

is the origin) and the area of triangle

OA_iA_{i+1}

is equal to

\frac{1}{2}

for

i=0,1,\ldots,n-1

. Let

(B_0,B_1,\cdots,B_n)

be a sequence of points. We may insert a point

B

between

B_i

and

B_{i+1}

\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}

, and the resulting sequence

(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)

is called an extension of the original sequence. Given two interesting sequences

(C_0,C_1,\ldots,C_n)

and

(D_0,D_1,\ldots,D_m)

, prove that if

C_0=D_0

and

C_n=D_m

, then we may perform finitely many extensions on each sequence until the resulting two sequences become identical.

Problem Statement