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0
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Let $$a,b,c,$$ and $$d$$ be distinct real numbers such that

\begin{align*} a &= \sqrt{4 + \sqrt{5 + a}}, \\ b &= \sqrt{4 - \sqrt{5 + b}}, \\ c &= \sqrt{4 + \sqrt{5 - c}}, \\ d &= \sqrt{4 - \sqrt{5 - d}}. \end{align*}

Compute $$abcd.$$

Sep 1, 2020

#3
+1038
+3

For anyone who can't seem to see the LaTeX, here is the picture:

(I plugged it into a latex site and screenshoted it)

But I'm not just here to give you a picture of something you probably already know.

After quite a long time of confusion, I finally got the answer! I actually first searched this question just to make sure that it wasn't answered before. And I found this link. Guest's answer confused me, because he/she said there was no answer. So I decided to try myself, and I got an answer to an "unanswerable" question.

Let the variable 'x' represent a, b, c and d.

We can say this:

Square both sides:

Subtract 4:

Square again!

Multiply out the left side:

Subtract 5:

We could keep on going, but remeber, we want abcd, not the equation for it. Using vieta's formula, we see that abcd = 11/1 = 11

:)

Sep 1, 2020
#4
+1

Ohhhh I see this makes so much sense! Thank you so much!!!

Guest Sep 1, 2020
#5
+111547
+1

Thanks ilorty,

It took a little while for me to get my head around what you have done.

It is an impressively simple answer.

Melody  Sep 1, 2020
#6
+1038
+2

Thanks! Maybe I'll practice my explaination skills, so that it won't be that hard to understand my answers....

ilorty  Sep 1, 2020
#7
+111547
+2

It wasn't your explanation at fault, it was my brain.  LOL

Melody  Sep 1, 2020