2 Equalities in Different Veins

by agbdmrbirdyface, Aug 25, 2016, 12:37 AM

Both of these problems were problems I solved way back in spring for a math camp during spring break (we don't question why).
Areteem Institute, Algebraic Transformations wrote:
Given that $abcd = 1$, show that
$\frac{a}{abc + ab + a + 1} + \frac{b}{bcd + bc + b + 1} + \frac{c}{cda + cd + c + 1} + \frac{d}{dab + da + d + 1} = 1$

Solution:
Areteem Institute, Algebraic Transformations wrote:
Given that $1991x^3 = 1992y^3 = 1993z^3 $ and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$, prove that $\sqrt[3]{1991} + \sqrt[3]{1992} + \sqrt[3]{1993} = \sqrt[3]{1991x^2 + 1992y^2 + 1993z^2}$

Solution:

Comment

5 Comments

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Wait that was pr0

Just like you
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:11 AM

by skipiano, Aug 25, 2016, 1:08 AM

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Oh wait these problems

I think I saw them among others when I went to MathZoom back in 2012!

Good to see camps are recycling problems!
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:13 AM

by djmathman, Aug 25, 2016, 1:20 AM

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wait MathZoom/Areteem Institute had a location in my area this year for spring break
so I took advantage of it and I went for a week
it was pretty decent

Nice
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:14 AM

by agbdmrbirdyface, Aug 25, 2016, 2:37 AM

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sorry for double posting but
Motivation

Nice
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 7:15 AM

by agbdmrbirdyface, Aug 25, 2016, 2:40 AM

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these are pretty problems

3pretty5you
This post has been edited 1 time. Last edited by shiningsunnyday, Aug 25, 2016, 5:03 PM

by cjquines0, Aug 25, 2016, 3:08 PM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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