Linear Equation and Inequality

Algebra Level 5

$\begin{aligned} \sqrt{55} < && \sqrt{(a+5)^2 + (b+4)^2 + (c+3)^2 + (d+2)^2 +(e+1)^2} \\ &-& \sqrt{a^2 + b^2 + c^2 + d^2 + e^2} \end{aligned}$

Suppose that there exist real numbers $a,b,c,d$ and $e$ such that $a+2b+3c+4d+5e=0$ and fulfill inequality above, find the value of $5a+4b+3c+2d+e$ .

12 15 None of these choices 13 14 Infinitely Many Value

1 solution

Paul Ryan Longhas
Jul 20, 2015

Let $\vec{L} = (a, b, c, d, e)$ and $\vec{M} = (5,4,3,2,1)$

So, the inequality implies that $|| \vec{L} + \vec{M} || - ||\vec{L}|| > || \vec{M}||$ .

Thus, $|| \vec{L} + \vec{M} || > ||\vec{L}|| + || \vec{M}||$ which is a contradiction of Triangle Inequality Theorem.

Therefore, $5a + 4b + 3c + 2d + e$ has no Real Value. Thus, the answer is None of the choices .

Is there any other way? cause I generally don't know the vector things.

Shishir Shahi - 3 years, 10 months ago

Nice tricky question.

Ayush Verma - 5 years, 10 months ago