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 the smallest is $b$
• B. The largest is $a$ and the smallest is $c$
• C. The largest is $c$ and the smallest is $e$
• D. The largest is $c$ and the smallest is $b$

Answer 1

The correct option is A The largest is $a$ and the smallest is $b$

We have, $d+e<a+b<c+d\Rightarrow d+e<c+d\Rightarrow e<c$

$\Rightarrow$$b+c<d+e<a+b\Rightarrow b+c<a+b\Rightarrow c<a$

$\Rightarrow$$a+b<c+d$ we know, $c<a$, then $d$ must be greater then $b\Rightarrow b<d$

$\Rightarrow$$d+e<a+b$ and $b<d\Rightarrow e<a$

$\Rightarrow$$c+d<e+a$ and $e<c\Rightarrow d<a$

and $a+b<c+d$ and $d<a\Rightarrow b<c$

Thus, we get $a$ is largest and $b$ is smallest.