Question

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
(A) 52 (B) 64 (C) 69 (D) 73

Answer 1

x=3, y=3, z=4

Answer 2

how could you get the values?Can u do The process?

Answer 3

using geometry, xyz is the volume of a rectangular prism, a cube has maximal volume
similarly, xy+yz+zx is half the surface area of a rectangular prism, a cube has maximal surface area
since we want the shape to be as close to a cube as possible, we want x,y,z values to be as close to each other as possible. this occurs when x=3,y=3,z=4

if you've learned calculus, i can show you a calculus solution to this problem

Answer 4

Yes,i know calculus.Kindly,show me the calculus method.

Answer 5

I agree with @eashy the answer is 69 when x=3, y=3 and z=4

Answer 6

http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx might help

Answer 7

You really cannot us Lagrange Multipliers. This function is discrete.

Answer 8

There are only few (x,y,z) positive and integers so that x+y+z=10. Evaluate the expression in each of them and conclude

Answer 9

Yes,i also think so.Can u give any other process@eliassaab

Answer 10

You have 8 possibilities for x , y and z to sum up to 10. Notice that your expression is symmetric with respect to x, y and z
Here they are
{{1, 1, 8}, {1, 2, 7}, {1, 3, 6}, {1, 4, 5}, {2, 3, 5}, {2, 4, 4}, {2,
6, 2} , {3, 3, 4}}
Test all of them you find that the (3,3,4) gives you a maximum

Answer 11

Any other process without hit and trial.

Answer 12

you have a lot of possibilities indeed

Answer 13

what if you have x = 10 y = 0 z = 0!?

Answer 14

@UsukiDoll,then it would become 0.

Answer 15

you can use Lagrange Multipliers to find the maximum value at x=y=z=3.33, then evaluate the expression to get 70.1927 which is less than D, then start testing integer values such that the average of x,y,z and also each x,y,z value are as close to 3.33 as possible (x=3, y=3, z=4), at which point you get answer C and are done, and only need to test one pair.

Answer 16

Here are the values of your expressions at the eight possibilities
{25, 37, 45, 49, 61, 64, 52, 69}

Answer 17

This problem can be done without any calculus and can be explained to students with little background in math. One should not drive a car to go across the street to visit the neighbor. One should walk there.

Answer 18

@ganeshie8 @satellite73 @experimentX

Answer 19

You meant to say that hit and trial is the easiest way to do this sum.

Answer 20

I would never ever attempt trial and error...too much time

Answer 21

@sagnikkarkar little bit of trial and error is necessary for these problems. Knowing how to use symmetry and AM-GM inequality makes u decide fast. Eashy's first solution is looking good to me. It requires you to try the middle values -- you can never avoid that with any other method I feel...
Also, you dont have a hundred cases. There are only 8 cases here. If you use symmetry, you will be looking up for maximum value of xyz only... so working out all the cases exhaustively is not as bad as it seems...

Answer 22

Can this problem be done by am-gm inequality?

Answer 23

Yes, I have tried AM-GM... that method also requires u to trial and error in the end... It is not giving out integer solutions :


GM <= AM
xyz <= (10/3)^3
<= 37

this is maximum when x=y=z=3.33

Answer 24

you're doing this problem for fun... or is this a homework problem where you need to show some work ?

Answer 25

Actually,i have to show some work,and i am trying this since 2 months,but cannot get any definite process.

Answer 26

Oh ok.. when i googled, i found some legitimate solution... one sec let me search for the site again ...

Answer 27

first problem :
http://www.resonance.ac.in/reso/downloads/KVPY-2013/KVPY-2013-SA-Solution.pdf

Answer 28

see if the solution there looks any convincing....
To me, it doesnt look any better than the first solution provided by Eashy...

Answer 29

Is it too difficult to sort out what is the largest of 8 numbers?