Weird sum

Algebra Level 5

Let f ( a , b , c ) = 1 1 + a + a b + a b c f (a, b, c)=\frac {1}{1+a+ab+abc} If w , x , y , z w, x, y, z are real numbers such that w x y z = 1 wxyz=1 , let M M and m m be the maximum and minimum values of f ( w , x , y ) + f ( x , y , z ) + f ( y , z , w ) + f ( z , w , x ) f (w, x, y)+f (x, y, z)+f (y, z, w)+f (z, w, x) respectively. Find M − m M-m .


The answer is 0.

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Joel Tan by Joel Tan , 21, Singapore

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2 solutions

Rajen Kapur
Apr 8, 2015

One interesting substitution to solve this easily is by putting w = p/q, x = q/r, y = r/s and z = s/p where p, q, r, and s are real. Getting a symmetric expression for the function makes it easy to conclude that the sum of four the functions is indeed equal to 1.

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Substitute 1 by x y z w xyzw and the whole expression simplifies to 1.

PS - if you cannot simplify,feel free to ask in comments section.

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