1. ## Re: GLaw’s challenge problems

Spoiler:
$a,\ b,\ c,\ x,\ y\ are\ all\ real\ and \ge 0.$

$STEP\ I:$

$0 \le \sqrt{xy} \le \dfrac{x + y}{2} \implies 0 \le \dfrac{x + y}{2} - \sqrt{xy} \implies$

$0 \le \left(\dfrac{x + y}{2} - \sqrt{xy}\right)^2 = \left(\dfrac{x + y}{2}\right)^2 - 2\sqrt{xy}\left(\dfrac{x + y}{2}\right) + xy \implies$

$0 \le \left(\dfrac{x + y}{2}\right)^2 - (x + y)\sqrt{xy}+ xy \implies$

$(x + y)\sqrt{xy} \le \left(\dfrac{x + y}{2}\right)^2 + xy.$

$STEP\ II:$

$\dfrac{a^2 + b^2 + c^2 + 3(ab + bc + ac)}{2} = \dfrac{2a^2 + 2b^2 + 2c^2 + 6(ab + bc + ac)}{4} =$

$\dfrac{(a^2 + 2ab + b^2) + (a^2 + 2ac + c^2) + (b^2 + 2bc + c^2) + 4(ab + bc + ac)}{4} =$

$\left\{\left(\dfrac{a + b}{2}\right)^2 + ab\right\} + \left\{\left(\dfrac{a + c}{2}\right)^2 + ac\right\} +\left\{\left(\dfrac{b + c}{2}\right)^2 + bc\right\}.$

$STEP\ III:$

$By\ STEP\ I:$

$(a + b)\sqrt{ab} \le \left(\dfrac{a + b}{2}\right)^2 + ab,\ and$

$(a + c)\sqrt{ac} \le \left(\dfrac{a + c}{2}\right)^2 + ac,\ and$

$(b + c)\sqrt{bc} \le \left(\dfrac{b + c}{2}\right)^2 + bc.$

$STEP\ IV:$

$Combining\ STEPS\ II\ and\ III:$

$(a + b)\sqrt{ab} + (a + c)\sqrt{ac} + (b + c)\sqrt{bc} \le \dfrac{a^2 + b^2 + c^2 + 3(ab + bc + ac)}{2}.$

I had the general idea right away, but finding the specific route took a while. Thanks for the challenge.

I still think you should put a new challenge into a new thread.

2. ## Re: GLaw’s challenge problems

Thanks, Jeff.

Originally Posted by JeffM
I still think you should put a new challenge into a new thread.
I’m out of challenge problems for now but if I have a new one I’ll start a new thread.

Solution to Problem #5:
Spoiler:
Expand $\left(\sqrt a-\sqrt b\right)^4+\left(\sqrt b-\sqrt c\right)^4+\left(\sqrt c-\sqrt a\right)^4\geq0$.

3. ## Re: GLaw’s challenge problems

You can borrow this one:
why is number 18 a good pick for me right now?

4. ## Re: GLaw’s challenge problems

Originally Posted by DenisB
You can borrow this one:
why is number 18 a good pick for me right now?
Oh! Is your girlfriend finally legal now?

-Dan

5. ## Re: GLaw’s challenge problems

Originally Posted by topsquark
Oh! Is your girlfriend finally legal now?
Dan me good man, when you made that post,
19 would have been a good pick for you: why?

6. ## Re: GLaw’s challenge problems

Originally Posted by DenisB
Dan me good man, when you made that post,
19 would have been a good pick for you: why?
Oh, I don't know, I can think of about 20 reasons.

-Dan

7. ## Re: GLaw’s challenge problems

Because it was post#19 !!

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