2004 Pan African MO Problems/Problem 4
Three real numbers satisfy the following statements:
(1) the square of their sum equals to the sum their squares.
(2) the product of the first two numbers is equal to the square of the third number.
Find these numbers.
Let be the three real numbers, and from the initial statements, we know that and .
Expanding, substituting, and simplifying results in By the Zero Product Property, either or .
If , then . Since , , and , we must have in this case. Substituting the values back in satisfies both of the original equations.
If , then , so by the Zero Product Property, or . If , then we have , so any real value of can work. If , then we have , so any real value of can work. Therefore and are also solutions (where and are reals), and substituting the values back in satisfies both of the original equations.
Therefore, the numbers that satisfies the two statements are either all zeroes or two zeroes and one non-zero number.
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