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Let x, y, and z be positive real numbers. Find the minimum value of

 May 30, 2020
 #1
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-1

By Cauchy-Schwarz, the minimum value is (1 + 2 + 4)^2 = 49.

 May 30, 2020
 #2
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0

a=1; b=1;c=1;p=39;d=(a +2*b+ 4*c) *(4/a + 2/b + 1/c);if(d==36, goto6, goto7);printd, a, b, c;p=p-1;if(p>=1, goto4,0); a++;if(a<50, goto4, 0);a=1;b++;if(b<50, goto4, discard=0; a=1;b=1;c++;if(c<50, goto4, 0)

 

The minimum value =36 and x, y, z take on an infinite number of values. Here are just a few:

mv   x y z

36    4 2 1
36    8 4 2
36    12 6 3
36    16 8 4
36    20 10 5
36    24 12 6
36    28 14 7
36    32 16 8
36    36 18 9
36    40 20 10
36    44 22 11
36    48 24 12
 

 Jun 1, 2020

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