If a, b, c, d and e are real numbers such that a + b < c + d, b + c < d + e, c + d < e + a and d + e < a + b, then

a + b < c + d ..... (1)

b + c < d + e ...... (2)

c + d < e + a ....... (3)

d + e < a + b ....... (4)

From (1) and (4):

d + e < a + b < c + d => e < c

From (1) and (3):

a + b < c + d < e + a => b < e

From (2) and (4):

b + c < d + e < a + b => c < a

Now, 

b < e < c < a

Let b = 1, e = 2, c = 3 and a = 4

From (1) and (3): 

a + b < c + d => 4 + 1 < 3 + d => d > 2 => d > e

c + d < e + a => 3 + d < 2 + 4 => d < 3 => d < c

Hence, 

b < e < d < c < a

Largest number is a and smallest number is b.