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Question

Let a,b,c,d,e be real numbers such that a+b<c+d, b+c<d+e, c+d<e+a, d+e<a+b. Then

A
The largest is a and tha smallest is b
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B
The largest is a and tha smallest is c
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C
The largest is c and tha smallest is e
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D
The largest is c and tha smallest is b
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Solution

The correct option is A The largest is a and tha smallest is b
a+b<c+d (1)b+c<d+e (2)c+d<e+a (3)d+e<a+b (4)

From (1) and (3)
a+b<e+ab<e (A)
From (2) and (4)
b+c<a+bc<a (B)

(1)(2), we get
ac<cec>e (C)

From (A),(B),(C), we get
b<e<c<a

(1)(4), we get
(ae)+(bd)<(ca)+(db)0<(ae)+(ac)<2(db)d>b

(1)+(3)(2), we get
c>d
a>c>(d,e)>b where d and e lies between a and b.
Hence a is the largest and b is the smallest.

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