By G. Hardy

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After the transformation is carried out k-times (k 2 l), we find that the numbers of red and black checkers are equal. Show that n is a multiple of 4. 8 The first author was the leader of the Chinese Olympiad team at the 29th IMO in July 1988 in Canberra, Australia. Problem pamphlets written in Russian were distributed as presents from the Soviet team. One of the problems was the following. Let a ' , a2, . . , a, be arbitrarily chosen real numbers. For another arbitrary real number a,form la1 - al, la2 - 011,.

Continuing thus, one obtains a decreasing sequence of integers r1, r2. . satisfying rj-2 = rj-lqj +rj. , rk-1 = rkqk+l. It is then readily verified that the GCD of a and b is d = rk. Indeed it is evident from the equations above that every common divisor of a and b divides rl , r2, . . , rk ; and moreover, viewing the equations in the reverse order, it is clear that rk divides each rj and hence also divides b and a . Euclid’s GCD algorithm can be viewed as an iteration of division with remainder.

Since both Tx and y contain an even number of l’s, the nth components of these vectors must also be the same. If we start with x1 = 1, we get the other preimage of y; it is related to the previous one by changing every 0 to 1 and every 1 to 0. Thus T has an inverse on X O ,and the lemma is proved. 6) Th’p~ = Th X. org/terms 8. CHECKERS ON A CIRCLE 37 We want to show that this equation holds also with h = 0. This follows from the fact that all our vectors are in the set XO where T has an inverse.