Absolute Value

Algebra Level pending

Given that n n real numbers x 1 , x 2 , . . . , x n x_1,x_2,...,x_n satisfy ∣ x i ∣ < 1 \mid x_i\mid<1 ( i = 1 , 2 , . . . , n i=1,2,...,n , and ∑ i = 1 n ∣ x i ∣ = 49 + ∣ ∑ i = 1 n x i ∣ \sum_{i=1}^n| x_i|=49+|\sum_{i=1}^nx_i| Find the minimum possible value of n n .


The answer is 50.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

LHS < n , RHS > 49 ⇒ n > 49 \mbox{LHS}<n,\mbox{RHS}>49\Rightarrow n>49

Now, we simply have to find an arrangement for n = 50 n=50 . Clearly, when x i = ( − 1 ) i ⋅ 49 50 x_i=(-1)^i\cdot\frac{49}{50} , the equation is satisfied. Hence, the answer is 50.

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