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I have to prove that for every non-zero integer $x,y,z$ at least one of the products $xy,xz,yz$ must be positive.

What I try to do is proof by contradiction. Assume $xy,xz,yz$ are all negative

$$xy = -a$$ $$xz = -b$$ $$yz = -c$$

Let's say:

$$(xy)(yz) = (-a)(-c) = ac$$

Is it possible to use transitivity for product of numbers to say that $$(xy)(yz) = xz$$ the equation above doesn't make sense yes I know. How else can I approach this problem. It's relatively easy but I just can't seem to prove that the assumption is wrong to get a contradiction..

Or can I just show a simple counter example to disprove the assumption.. so let

x be positive

y be negative and

z be negative.

From this, I can show y.z = negative . negative = positive. Hence, the assumption is false and we have a contradict.

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  • $\begingroup$ $x,y,z$ are nonzero....can they be negative? $\endgroup$
    – Soham
    Nov 10 '15 at 3:41
  • $\begingroup$ @tatan Sure; otherwise the problem would not make much sense. $\endgroup$
    – user147263
    Nov 10 '15 at 3:42
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    $\begingroup$ Overthinking. Four cases. (ii) all positive. Then any product of two is positive; (ii) two positive and one negative. Then the product of the two positives is positive; (iii) two negative and one positive. Then the product of the two negatives is positive; (iv) it's your turn. $\endgroup$ Nov 10 '15 at 3:43
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@Andre Nicolas' comment is natural but a little bit long to write. Here is another way, using contradiction.

If $xy$, $yz$ and $xz$ are all negative, then their product is also negative (product of 3 negative numbers).

But their product is $(xy)(yz)(xz)=(xyz)^2$, which is obviously positive. Therefore, we have a contradiction: at least one of the numbers is positive.

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Since the integers are non-zero, therefore either they are positive or negative. You have two buckets: one for positive integers and one for negative integers. You have $3$ objects to be distributed into two buckets, so one of the buckets will have at least two objects in it. So the product of the two objects from the same bucket has to be positive.

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By Pigeon-Hole-Principle we have,three variables $x,y,z$ which can be assigned two values -either positive or negative. So, at least one value (+ve or -ve) will have more than $$[\frac {3-1}{2}]=1$$ variables.The other will obviously have two values and we know product of two positive or negative numbers is always positive.

Note:-The word at least is of special significance.

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