Waste less time on Facebook — follow Brilliant.
×

Good Inequality Problem

Given real numbers \(a, b\) and \(c\) so that \(a^2 + b^2 + c^2 = k > 0.\) Then prove following inequality:

\[\frac{2}{k}abc - \sqrt{2k} \leq a + b + c \leq \frac{2}{k}abc + \sqrt{2k}\]

Note by Fariz Azmi Pratama
4 years, 3 months ago

No vote yet
22 votes

Comments

Sort by:

Top Newest

Here's my solution:

Without loss of generality, assume that \(|b|\leq |c|\leq |a| \). Then by \(Cauchy - Schwarz\) Inequaility we have

\[\begin{aligned} \left(a+b+c-\frac{2}{k}abc\right)^2 &= \left[a\left(1-\frac{2}{k}bc\right) + (b+c)\right]^2 \\ &\leq [a^2+(b+c)^2]\left[\left(1-\frac{2}{k}bc\right)^2 +1^2\right]\\ &= (k + 2bc) \left(2 - \frac{4}{k}bc +\frac{4}{k^2}b^2c^2\right) \\ &= 2k +\frac{8}{k^2} b^3c^3 - \frac{4}{k}b^2c^2\\ &= 2k +\frac{4}{k^2}b^2c^2(2bc - k) \end{aligned}\] Since \(|b|\leq |c|\leq |a| \), \(a^2 \geq\frac{k}{3}\), which implies that \[\begin{aligned} 2bc \leq b^2 + c^2 = k - a^2 \leq k - \frac{k}{3} \leq \frac{2k}{3}< k \end{aligned}\] Therefore, \(2bc - k < 0\), which yields \(\left(a+b+c-\frac{2}{k}abc\right)^2 \leq 2k\) or \[\begin{aligned} \frac{2}{k}abc - \sqrt{2k}\leq a+b+c\leq \frac{2}{k}abc +\sqrt{2k} \end{aligned}\]

Russelle Guadalupe - 4 years, 2 months ago

Log in to reply

Good solution. Your main observation is that the inequality is equivalent to showing that \[ \left[ (a+b+c)(a^2+b^2+c^2) - 2abc \right]^2 \leq 2 (a^2+b^2 + c^2 )^3. \]

This could also be done through brute force expansion and then AM-GM several times (i.e. Muirhead).

Calvin Lin Staff - 4 years, 2 months ago

Log in to reply

Great solution. Just to state that \(2bc \leq b^2+c^2 < k\) since \(a\) cannot be zero otherwise all of them are 0, which contradicts the condition \(k>0\), just showing this so we do not need the \(\frac{k}{3}\) thing.

Yong See Foo - 4 years, 2 months ago

Log in to reply

Hands down, your solution answered it all. Thanks :)

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

Hi, if we substitute \(k = a^2 + b^2 + c^2\) into the inequalities, then the system of inequalities is homogeneous, let \(u = a + b + c\), \(v = ab + bc + ca\) and \(w = abc\), then the system becomes \[\frac{2w}{\sqrt{u^2 - 2v}} - \sqrt{2u^2 - 4v} \leq u \leq \frac{2w}{u^2 - 2v} + \sqrt{2u^2 - 4v}\] i.e. \[\frac{2w}{\sqrt{u^2 - 2v}} - \sqrt{2u^2 - 4v} - u \leq 0 \leq \frac{2w}{u^2 - 2v} + \sqrt{2u^2 - 4v} - u\] Both expressions are linear in \(w\). By \(uvw\) method it suffices to prove inequality in two cases 1) \((a - b)(b - c)(c - a) = 0\), 2) \(abc = 0\), in the first case we let WLOG \(b = c\) and we have to prove \[\frac{2ab^2}{a^2 + 2b^2} - \sqrt{2a^2 + 4b^2} \leq a + 2b \leq \frac{2ab^2}{a^2 + 2b^2} + \sqrt{2a^2 + 4b^2}\] which is easy. In the second case WLOG let \(c = 0\) and we have to prove \[-\sqrt{2a^2 + 2b^2} \leq a + b \leq \sqrt{2a^2 + 2b^2}\] which is again easy.

Here is link to the method used: PDF I used different substitution, but it doesn't interfere with the method.

Jan J. - 4 years, 2 months ago

Log in to reply

It took me a bit longer to understand, but I got it. Thanks :)

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

for 3 real numbers a,b and c, notice that AM=(a+b+c)/3, GM=(abc)^(1/3) and RMS=sqrt[(a^2+b^2+c^2)/3]. try to use RMS>AM>GM.i tried and it worked.but the solution is around 20-22 lines long.so will take a long time to type :-(

Adeeb Zaman - 4 years, 2 months ago

Log in to reply

How do you account for the fact that a,b, and/or c could be negative numbers too. I thought AM-GM only works for positive numbers.

Edwin Ma - 4 years, 2 months ago

Log in to reply

Wow that is so long. You did it great. But I think for this problem, \(AM - GM\) has nothing to do here. Like what David say, \(AM - GM\) only works on positive number. Let say for your case \(a = 1, b = -1\) and \(c = 0\) then \(AM = 0 < GM = \sqrt[3]{2}.\)

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

Any chance you could tell us where you got this problem?

Sotiri Komissopoulos - 4 years, 2 months ago

Log in to reply

Oh sorry, my friend gave me this problem. Unfortunately she didn't mention where she get this problem from.

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

That's fine, thanks. :)

Sotiri Komissopoulos - 4 years, 2 months ago

Log in to reply

Clearly the leftmost side is less than the rightmost side as \(k\) is positive. But beyond that, I'll have to do more than a first time glance to get this!

Tanishq Aggarwal - 4 years, 2 months ago

Log in to reply

Yes, and to be honest I don't have any clue to solve this inequality.

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

Put the value of \(k\) in both sides, square up, rearrange & you will get it after some reasoning...

Paramjit Singh - 4 years, 2 months ago

Log in to reply

What do you mean by put the value of \(k\) in both sides? Since there are three sides on the ineq.

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

easy

Youssef Ihizem - 4 years, 2 months ago

Log in to reply

Would you like to show your solution or maybe some hints?

Fariz Azmi Pratama - 4 years, 2 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...