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Print49th Mathematical Olympiad in Ukraine
Ukraine algebra
Problem
Distinct real numbers satisfy the condition . Find all possible values of the product : 1) for all real ; 2) for positive real ?
Solution
a) See problem 9.5
b) We will show that there are no such positive numbers that satisfy the condition of the problem. Without loss of generality we suppose that , then from the equality we conclude that . From the condition , we can get a contradiction: .
b) We will show that there are no such positive numbers that satisfy the condition of the problem. Without loss of generality we suppose that , then from the equality we conclude that . From the condition , we can get a contradiction: .
Final answer
For real a, b, c: the product can be −1 or 1. For positive a, b, c: no such triples exist, so there are no possible values.
Techniques
Simple EquationsOther