Find one solution for system of inequalities (if exists) If there exists a solution for following s

hughy46u

hughy46u

Answered question

2022-05-28

Find one solution for system of inequalities (if exists)
If there exists a solution for following system of inequalities, find at least one, else show that there is no solution exists.
x 4 ( x 1 + x 7 ) > x 3 ( x 2 + x 8 )
x 3 ( x 5 + x 8 ) > x 7 ( x 4 + x 1 )
x 7 ( x 6 + x 4 ) > x 8 ( x 5 + x 3 )
x 8 ( x 3 + x 2 ) > x 4 ( x 6 + x 7 )
for all i, 0 < x i < 1 and i x i = 1

Answer & Explanation

Alessandro Schmidt

Alessandro Schmidt

Beginner2022-05-29Added 6 answers

A quick computer simulation shows such solutions exist. Example:
A = (0.020302010304655, 0.052938093317538, 0.040635876493691, 0.016599197409892, 0.247812843241618, 0.120692838614077, 0.388175655457951, 0.112843485160578)
sum(A)
ans=1
A ( 4 ) ( A ( 1 ) + A ( 7 ) ) A ( 3 ) ( A ( 2 ) + A ( 8 ) )
ans = 4.372166355951553e-05
A ( 3 ) ( A ( 5 ) + A ( 8 ) ) A ( 7 ) ( A ( 4 ) + A ( 1 ) )
ans = 3.314355258354201e-04
A ( 7 ) ( A ( 6 ) + A ( 4 ) ) A ( 8 ) ( A ( 5 ) + A ( 3 ) )
ans = 0.020743867247722
A ( 8 ) ( A ( 3 ) + A ( 2 ) ) A ( 4 ) ( A ( 6 ) + A ( 7 ) )
ans = 0.002112404285034
groupweird40

groupweird40

Beginner2022-05-30Added 1 answers

Converting the elements of A to rationals and adding those, one obtains
112204726358063191347750231121831481 112204726389776842878359836410752160 ,
not 1.

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