IMO 1967 LL USS57

Origin: USS

Problem

Consider the sequence (cn): c1 = a1 + a2 + \cdot \cdot \cdot + a8, c2 = a2 1 + a2 2 + \cdot \cdot \cdot + a2 8, . . . . . . . . . . . . cn = an 1 + an 2 + \cdot \cdot \cdot + an 8, . . . . . . . . . . . . 2 This problem is not elementary. The solution offered by the proposer, which is not quite clear and complete, only shows that if such a \beta exists, then \beta \geq 2(1−\alpha). where a1, a2, . . . , a8 are real numbers, not all equal to zero. Given that among the numbers of the sequence (cn) there are infinitely many equal to zero, determine all the values of n for which cn = 0.

Solution

Obviously cn > 0 for all even n. Thus cn = 0 is possible only for an odd n. Let us assume a1 \leqa2 \leq\cdot \cdot \cdot \leqa8: in particular, a1 \leq0 \leqa8. If |a1| < |a8|, then there exists n0 such that for every odd n > n0, 7|a1|n < an 8 ⇒an 1 + \cdot \cdot \cdot + an 7 + an 8 > 7an 1 + an 8 > 0, contradicting the condition that cn = 0 for infinitely many n. Similarly |a1| > |a8| is impossible, and we conclude that a1 = −a8. Continuing in the same manner we can show that a2 = −a7, a3 = −a6 and a4 = −a5. Hence cn = 0 for every odd n.