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Anybody got some challenging problems? (algebra preferred) Anything below calculus. 

 Nov 3, 2019
 #1
avatar+94 
+1

Find the smallest possible value of

\(\frac{(y-x)^2}{(y-z)(z-x)} + \frac{(z-y)^2}{(z-x)(x-y)} + \frac{(x-z)^2}{(x-y)(y-z)}, \)

where x, y and z are distinct real numbers.

 Nov 3, 2019
 #2
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min{(y - x)^2/((y - z) (z - x)) + (z - y)^2/((z - x) (x - y)) + (x - z)^2/((x - y) (y - z))} = 3 

 

at (x, y, z) = (33/10, 8/5, -23/5)

 Nov 3, 2019
 #3
avatar+94 
+1

Correct!

VooFIX  Nov 3, 2019
 #4
avatar+94 
+1

Find the common ratio of the infinite geometric series

\(\frac{5}{6}-\frac{4}{9}+\frac{32}{135}-\dots\)

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 Nov 3, 2019
 #5
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0

Common ratio = 8 / 15

 Nov 3, 2019
 #6
avatar+94 
+1

The answer 8/15 is incorrect. The correct answer is \(-8/15\)

VooFIX  Nov 3, 2019
edited by VooFIX  Nov 3, 2019

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