Suppose we know that (a+b+d)/(a'+b'+d')<= M and (a+c+d)/(a'+c'+d')<= M and (a)/(a')<=(b)/(b')<=(d)/(d') and (a)/(a')<=(c)/(c')<=(d)/(d') and also that a,b,c,d,a′,b′,c′,d′ in (0,1]. Is the following true? (a+b+c+d)/(a'+b'+c'+d')<= M

Ishaan Booker 2022-07-21 Answered
Fraction inequality
Suppose we know that
a + b + d a + b + d M  and  a + c + d a + c + d M
and
a a b b d d  and  a a c c d d
and also that a , b , c , d , a , b , c , d ( 0 , 1 ]
Is the following true?
a + b + c + d a + b + c + d M
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Answers (1)

frisiao
Answered 2022-07-22 Author has 13 answers
No. Take all numbers equal to 1 2 , except for a , which is equal to 1. Besides, take M = 3 4 . Then
a + b + c + d a + b + c + d = 4 5 .

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