Algebra · Symmetric functions · Algebraic identities · Vieta relations · Self reciprocal polynomial · Products equal to one
Problem 3, 2013
Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(abcd = 1\) and
\[ a + b + c + d = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} . \]
Prove that some two of the numbers \(ab\), \(ac\), \(ad\), \(bc\), \(bd\), \(cd\) are equal.
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Serbian National Competition (Drzavno takmicenje) 2013, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source