The problem of the 100 hundred numbers.

Algebra Level 3

The numbers x 1 , x 2 , x 3 , . . . , x 100 x_1, x_2, x_3, ..., x_{100} satisfy that ∑ 1 ≤ i < j ≤ 100 ( i x j − j x i ) 2 = 338350 \sum_{1\leq i < j\leq 100}{(ix_j-jx_i)^2}=338350 and ∑ i = 1 100 x i 2 = 1. \sum_{i=1}^{100} {x_i}^2=1. Find ∑ i = 1 100 i x i \sum_{i=1}^{100}i x_i


The answer is 0.

Arturo Presa by Arturo Presa , 62, USA

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1 solution

Arturo Presa
Feb 10, 2018

Using Lagrange Identity , we obtain that ∑ 1 ≤ i < i ≤ 100 ( i x j − j x i ) 2 = ∑ i = 1 100 i 2 ∑ i = 1 100 x i 2 − ( ∑ i = 1 100 i x i ) 2 \sum_{1\leq i < i\leq 100}{(ix_j-jx_i)^2}=\sum_{i=1}^{100} i^2 \sum_{i=1}^{100}{ x_i} ^2- (\sum_{i=1}^{100}ix_i)^2

Then 338350 = 338350 − ( ∑ i = 1 100 i x i ) 2 , 338350= 338350-(\sum_{i=1}^{100}ix_i)^2, and, therefore, ∑ i = 1 100 i x i = 0 . \sum_{i=1}^{100}ix_i=\boxed{0}.

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