Show that the product of two of the numbers $(65^{1000} – 8^{2001} + 3^{177}), (79^{1212} – 9^{2399} + 2^{2001})$ and $(24^{4493} – 5^{8192} + 7^{1777})$ is non-negative, without actually evaluating the numbers.
P.S. I have found by calculation that all the three numbers are positive, but that does not solve the problem of proving without calculation.
Thanks in advance.
Best Answer
I think the trick is that the question just asks for a proof that there exist two of the numbers such that their product is non-negative. In principle there could be other products that were negative.
Now, if at least two of the numbers are non-negative, then their product is non-negative too.
If less than two of them are non-negative, there must be at least two negative numbers among them ...