Giving x 1 , x 2 , . . . . . . . . . x 2 0 1 6 real numbers in the interval [ − 1 , 1 ] such that x 1 3 + x 2 3 + . . . . . . . . . + x 2 0 1 6 3 = 0 .
Which is the greatest value of x 1 + x 2 + . . . . . . . . . + x 2 0 1 6 ?
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Up voted. Can any one explain why 1/2 is chosen ? Thank you.
x i = s i n ( α i ) .
s i n ( 3 γ ) = 3 s i n ( γ ) − 4 s i n ( γ ) 3
Summing all α i we get i = 1 ∑ n s i n ( 3 α i ) = 3 i = 1 ∑ n s i n ( α i ) − 4 i = 1 ∑ n s i n ( α i ) 3 = 3 i = 1 ∑ n x i − 4 i = 1 ∑ n x i 3 = 3 i = 1 ∑ n x i
s i n ( 3 γ ) ≤ 1 ⇒ 3 i = 1 ∑ n x i ≤ n and in this case, for n = 2 0 1 6 the greatest number is 3 n = 3 2 0 1 6 = 6 7 2
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