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## Homework Statement

Find A:

A

^{2}= (4/5)(1829

^{2}+ 1298

^{2})

in about 3min, because this comes at the end of a rather difficult geometry problem with 6 min for the entire question. (edit: Yes, calculators weren't allowed because it was a competition. I have verified that everyone comes to this step, so I didn't make any mistakes before that.)

## Homework Equations

None.

## The Attempt at a Solution

I'm quite sure there's some algebraic manipulation to help you with this one, but I can't think of it. I did the 1829

^{2}and 1298

^{2}the 'brute force' way, then added them together for the sum. This step took the longest - more than 3min for me.

Then I converted this to standard form so as to mentally estimate the common logarithm to 3 decimal points. I also know common lg 4 and lg 5 by memory to 3 decimal points - then I took a 'guess' at A:

lg A = (lg 4 + lg(1829

^{2}+1298

^{2}) - lg 5)/2

Comes around to 2 * 10

^{3}~ rather quickly, correct a bit and 'try' a few values, and I found 2006 eventually (edit: , which is the correct answer.)

Also did think of putting it in (1829+1298)

^{2}- 2*1829*1298, but it still takes about as long. Alternatively, I had a suggestion to use 1800

^{2}+ 29

^{2}+ 2*1800*29, and similar for 1200

^{2}+ 98

^{2}+ 2*1200*98, which takes about as long for me.

Does anyone see an algebraic manipulation to make this easier? Thanks in advance ;)

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