Pls refer to the attached picture for the question. On a side note, how do you enter avsolute value into the text box here? Thanks.

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Find the minimum n such that `|x_1|+|x_2|+ *** +|x_n|=49+|x_1+x_2+ *** +x_n| ` where ` ``x_i<1 ` for all i=1,2,...,n :

The right hand side of the equation is always greater than or equal to 49, so the left hand side must also be greater than or equal to 49.

If `x_i=1,i=1,2,...,n ` then you could have 49 terms; however each `x_i<1 ` so there must be more than 49 terms.

Claim: n=50 is the minimum.

(1) n cannot be 49 or smaller by the argument above.

(2) `x_i=(-1)^n(48/50),1<=i<=48, x_49=-1/2,x_50=1/2 `

Then `sum_(i=1)^n |x_i|=49 ` so the left hand side is 49.

Also `49+| sum_(i=1)^48 (-1)^i -1/2+1/2|=49+0=49 `

Therefore there exists 50 values for `x_i ` such that the equality holds.

Sources:
rimmery's profile pic

rimmery | Student, Undergraduate | (Level 1) Honors

Posted on

Just to answer the side question,

Absolute values can be created with vertical bars (inside the math backquotes of course)

Vertical bar is shift+backslash (the key above enter)

So `|x|` should look something like `|x|`

It automatically adjusts length/height according to the equation inside.

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