Prove that if real numbers a, b and c satisfy
a + b + c > 0, ab + ac + bc > 0 and abc > 0,
then each of a, b and c is positive.
To prove that each of the real numbers a, b, and c is positive, we will use a proof by contradiction.
Assume that at least one of the numbers a, b, and c is non-positive (i.e., negative or zero). Without loss of generality, let's assume that a ≤ 0.
Since a + b + c > 0, we can rearrange the inequality to b + c > -a.
Now, consider the second inequality, ab + ac + bc > 0. Since a ≤ 0, we can rewrite this inequality as b(a + c) > -ac.
Since a ≤ 0 and b + c > -a, we can multiply these two inequalities together to get b(a + c)(b + c) > -a(-ac), which simplifies to (b + c)^2 > ac.
Now, consider the third inequality, abc > 0. Since a ≤ 0 and b + c > -a, we can multiply these two inequalities by b to get ab(b + c) > -a(bc). Since abc > 0, we know that both ab(b + c) and -a(bc) have the same sign. Therefore, we can rewrite this inequality as ab(b + c) > -a(bc) > 0.
Now, let's combine the two inequalities we obtained: (b + c)^2 > ac and ab(b + c) > -a(bc) > 0.
Expanding the square on the left side of the first inequality, we have b^2 + 2bc + c^2 > ac. Now, subtracting ac from both sides, we get b^2 + 2bc + c^2 - ac > 0.
Adding this inequality to the second inequality we obtained, we get b^2 + 2bc + c^2 - ac + ab(b + c) > 0.
Factoring out the common term (b + c) from the first three terms, we have (b + c)(b + c - a) + ab(b + c) > 0.
Since a ≤ 0, b + c > -a. Now, since a ≤ 0, -a ≥ 0, which means -a(b + c) ≥ 0.
So, we can rewrite the inequality as (b + c)(b + c - a - ab) > 0.
Now, let's consider the two factors: (b + c) and (b + c - a - ab).
Since a ≤ 0 and b + c > -a, we know that b + c - a - ab > b + c - (-a) - 0 = b + c + a > 0.
Therefore, both factors are positive.
However, if the product of two factors is positive, then both factors must be positive. This contradicts our initial assumption that at least one of the numbers a, b, and c is non-positive.
Therefore, our assumption is false, and it follows that each of the real numbers a, b, and c must be positive.