Question

Five real numbers $x_1,x_2,x_3,x_4,x_5$ are such that: $\sqrt{x_1-1}+2\sqrt{x_2-4}+3\sqrt{x_3-9}+4\sqrt{x_4-16}+5\sqrt{x_5-25}=\frac{x_1+x_2+x_3+x_4+x_5}{2}$
The value of $\frac{x_1+x_2+x_3+x_4+x_5}{2}$ is

• A. Not uniquely determined
• B. $55$
• C. $110$
• D. $210$

Answer 1

Answer: (B)

$55$

$(x_1-2\sqrt{x_1-1})+(x_2-4\sqrt{x_2-4})+(x_3-6\sqrt{x_3-9})+(x_4-8\sqrt{x_4-16})+(x_5-8\sqrt{x_5-25})=0$
$\Rightarrow (\sqrt{x_1-1}-1{)}^2+(\sqrt{x_2-4}-2{)}^2+(\sqrt{x_3-9}-3{)}^2+(\sqrt{x_4-16}-4{)}^2+(\sqrt{x_5-25}-5{)}^2$
Now, $\sqrt{x_1-1}-1=0,\sqrt{x_2-4}-2=0,\sqrt{x_3-9}-3=0,\sqrt{x_4-16}-4,\sqrt{x_5-25}-5=0$
$x_1=2,x_2=8,x_3=18,x_4=32,x_5=50$
$\therefore \frac{x_1+x_2+x_3+x_4+x_5}{2}=55$