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Topic: inequality & constraints: x_i x_j <= -1/n (Read 592 times) |
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william wu
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inequality & constraints: x_i x_j <= -1/n
« on: Aug 23rd, 2003, 10:05pm » |
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Suppose you have n real numbers x1, x2, ... , xn that satisfy the following two conditions:
[sum]i=1 to n xi = 0
[sum]i=1 to n xi[sup2] = 1
Prove that there exists integers i and j between 1 and n such that xixj [le] -1/n.
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| « Last Edit: Aug 23rd, 2003, 10:06pm by william wu » |
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Pietro K.C.
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Re: inequality & constraints: x_i x_j <= -1/n
« Reply #1 on: Sep 3rd, 2003, 5:02pm » |
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Hint 1:
Assume the opposite: that xixj is greater than -1/n for all i, j.
Hint 2 (bigger):
Now you have constraints on the terms xixj and xi2. In what kind of mathematical thing do these two often appear together?
Hint 3 (spoiler):
If the sum of the xi is zero, so is the square of that sum. Work from there to derive a contradiction between that and your assumption that xixj is greater than -1/n for all i, j.
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| « Last Edit: Sep 3rd, 2003, 5:06pm by Pietro K.C. » |
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