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Let x, y, z be three non-negative integers such that x + y + z = 10. The maximum possible value of xyz + xy + yz + zx is

Let x, y, z be three non-negative integers such that x + y + z = 10. The maximum possible value of xyz + xy + yz + zx is

Grade:12

1 Answers

Piyush Kumar Behera
417 Points
7 years ago
so we can have max value at (4,4,(13-(4+4))i.e. 4,4,5So Maximum value = 4*4*5 - 11 = 69Now greatest integer function[13/3] = 4 Rule to get the maximum value, which we can get when the value of x+1,y+1 & z+1 is as near as possible AM-GMWe will now use(x+1) + (y+1)+ (z+1)=13 Now x+y+z=10 can be written as (x + 1)(y + 1)(z + 1) - 11  So now we have to maximizexyz+xy+xz+yz = (x + 1)(y + 1)(z + 1) - 11As x+y+z=10xyz+xy+xz+yz can be written as (x + 1)(y + 1)(z + 1) - (x + y + z + 1)We can easily say that
 
 
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