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

Answer 1

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

$a+b<c+d\cdots \left(1\right)b+c<d+e\cdots \left(2\right)c+d<e+a\cdots \left(3\right)d+e<a+b\cdots \left(4\right)$

From $\left(1\right)$ and $\left(3\right)$
$a+b<e+a\Rightarrow b<e\cdots \left(A\right)$
From $\left(2\right)$ and $\left(4\right)$
$b+c<a+b\Rightarrow c<a\cdots \left(B\right)$

$\left(1\right)-\left(2\right),$ we get
$a-c<c-e\Rightarrow c>e\cdots \left(C\right)$

From $\left(A\right),\left(B\right),\left(C\right),$ we get
$b<e<c<a$

$\left(1\right)-\left(4\right),$ we get
$(a-e)+(b-d)<(c-a)+(d-b)\Rightarrow 0<(a-e)+(a-c)<2(d-b)\Rightarrow d>b$

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