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a_{n+1}= |a_n - b_n|, b_{n+1}= |b_n - c_n|, c_{n+1}= |c_n - d_n|, d_{n+1}=...

Source: 1988 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

November 20, 2022
algebrarecurrence relation

Problem Statement

Starting with four given integers a1,b1,c1,d1a_1, b_1, c_1, d_1 is defined recursively for all positive integers nn: an+1:=∣an−bn∣,bn+1:=∣bn−cn∣,cn+1:=∣cn−dn∣,dn+1:=∣dn−an∣.a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|. Prove that there is a natural number kk such that all terms ak,bk,ck,dka_k, b_k, c_k, d_k take the value zero.
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