here's a riddle for you?

Question

only one of my friends can solve it (none of my friends are mathematical, though)

x + y + z =1

using only that fact, prove:
xy + yz + xz < 1/2

so far, Zorro has the only one that has proven the answer

so far, Zorro is the only one that has proven the answer*

and what are you talking about with the "none" nonsense?

FYI:
"With mass nouns, you have to use the singular. ("None of the
wheat is...") With count nouns, you can use either the singular or
the plural. ("None of the books is..." or "None of the books
are...") Usually, the plural sounds more natural, unless you're
trying to emphasize the idea of "not one", or if the words that
follow work better in the singular."

I believe that "friends" is a count noun. Correct me if I missed something!

http://www.yaelf.com/aueFAQ/mifnonevsnoneare.shtml

Answer 1

(x+y+z)(x+y+z) = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz = 1

(1-x^2 - y^2 - z^2) = 2xy + 2xz + 2yz

(1 - x^2 - y^2 - z^2)
------------------------ = xy + xz + yz
.............2

Clearly the left hand side is less than 1/2

Answer 2

x + y + z =1
(x + y + z)^2 =1^2
x^2 + y^2 + z^2 + 2xy + 2xz + 2yz = 1
2(xy + xz + yz) = 1 - x^2 + y^2 + z^2 < 1
xy + xz + yz = (1 - x^2 + y^2 + z^2)/2 < 1/2
xy + xz + yz < 1/2

Answer 3

None of your friends IS grammatical, either, evidently. "None" is treated as singular.

I don't know how Screamin got the correct answer, but you can use the "which one" method to find it. Let's say that a total higher than 1/3 could be achieved by making x or y or z larger, and one of the others smaller. Which one? How can a higher total be achieved by making x larger and y smaller? Wouldn't the same higher total be found by making y larger and x smaller? Since x, y, and z are interchangable, the highest total is as Screamin says, when all three are equal.

Answer 4

Actually, if your friend's really smart s/he'd point out that the maximal value of xy + yz + xz = 1/3 (the maximal value is when x = y = z)

Answer 5

I am just an oooooooolllllllllddddddddd lady. That makes my brain, if I have any left, tickle. Oh no, there is smoke coming out of my head. I told you not to make my brain think.

Answer 6

x= 0.33
y= 0.33
z= 0.33