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Part of 1976 Polish MO Finals

Problems(1)

a_{n+1} = a_n +b_n 4 recurrence relations

Source: Polish MO Finals 1976 p2

8/23/2024
Four sequences of real numbers (an),(bn),(cn),(dn)(a_n), (b_n), (c_n), (d_n) satisfy for all nn, an+1=an+bn,bn+1=bn+cn,a_{n+1} = a_n +b_n, b_{n+1} = b_n +c_n, cn+1=cn+dn,dn+1=dn+an.c_{n+1} = c_n +d_n, d_{n+1} = d_n +a_n. Prove that if ak+m=am,bk+m=bm,ck+m=cm,dk+m=dma_{k+m} = a_m, b_{k+m} = b_m, c_{k+m} = c_m, d_{k+m} = d_m for some k≥1,n≥1k\ge 1,n \ge 1, then a2=b2=c2=d2=0a_2 = b_2 = c_2 = d_2 = 0.
algebrarecurrence relation