1985 USAMO Problems/Problem 1
Determine whether or not there are any positive integral solutions of the simultaneous equations with distinct integers .
Lemma: For a positive integer , (Also known as Nicomachus's theorem)
Proof by induction: The identity holds for . Suppose the identity holds for a number . It is well known that the sum of first positive integers is . Thus its square is . Adding to this we get , which can be rewritten as This simplifies to . The induction is complete.
Let be the sum , and let be the sum . Then assign the value for each . Then:
Thus, a positive integral solution exists.
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