The positive numbers a, b, c, A, B, C satisfy
Prove that .
All Soviet Union Olympiad, 1987.
Proving that is equivalent to showing that
But this is equivalent to
If , then the above is obviously true. If not, then we have
and thus, our inequality holds.
Let a and b be distinct positive integers such that is divisible by . Prove that
Let and write and with . Then,
is an integer. N^ote that . Similarly, .
Because , we have
Now, we apply the following lemma: Let a and b be two coprime numbers. If c is an integer such that , then .
Hence, we get, implying that . Therefore,
It follows that . QED.
Note that the key step divides g can also be obtained by clever algebraic manipulations such as .
More Olympiad problems later,