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Part of 1990 Turkey Team Selection Test

Problems(1)

x_1 + x_2 + ... + x_n = 0

Source: Turkey TST 1990 - P2

9/11/2013
For real numbers xix_i, the statement x1+x2+x3=0⇒x1x2+x2x3+x3x1≤0 x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0 is always true. (Prove!) For which n≥4n\geq 4 integers, the statement x1+x2+⋯+xn=0⇒x1x2+x2x3+⋯+xn−1xn+xnx1≤0x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0 is always true. Justify your answer.
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