Proof of inequality

Question

I have problems with proving inequality :

$${a^{2}}+b^2+c^2+\frac{2}{5}abc<50$$

where $a,b,c$ are the lengths of triangle's sides, and the circumference of the triangle is $10$.

Thanks.

Answer 1

Consider the polynomial $(x-a)(x-b)(x-c)$. Multiplied out, this is

$$ \begin{eqnarray} (x-a)(x-b)(x-c) &=& x^3-(a+b+c)x^2+(ab+ac+bc)x-abc \\ &=& x^3-10x^2+(ab+ac+bc)x-abc\;. \end{eqnarray} $$

We also have

$$10^2=(a+b+c)^2=(a^2+b^2+c^2) + 2(ab+ac+bc)\;,$$ $$ab+ac+bc=\frac{100-(a^2+b^2+c^2)}{2}\;,$$

and thus

$$ (x-a)(x-b)(x-c) = x^3-10x^2+\frac{100-(a^2+b^2+c^2)}{2}x-abc\;. $$

Now since $a$, $b$ and $c$ form a triangle with perimeter $10$, they must all be less than $5$. Thus the value of the polynomial for $x=5$ is positive, that is,

$$ 5^3-10\cdot5^2+\frac{100-(a^2+b^2+c^2)}{2}\cdot5-abc>0\;, $$

which upon rearrangement becomes your inequality.