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5

Part of 1989 Romania Team Selection Test

Problems(1)

The number of latticial cycles

Source: Romania TST for BMO 1989 P5

2/21/2014
A laticial cycle of length nn is a sequence of lattice points (xk,yk)(x_k, y_k), k=0,1,⋯ ,nk = 0, 1,\cdots, n, such that (x0,y0)=(xn,yn)=(0,0)(x_0, y_0) = (x_n, y_n) = (0, 0) and ∣xk+1−xk∣+∣yk+1−yk∣=1|x_{k+1} -x_{k}|+|y_{k+1} - y_{k}| = 1 for each kk. Prove that for all nn, the number of latticial cycles of length nn is a perfect square.
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