The aim of this article is to present some interesting results gained on the basis of long-range working of both authors with mathematically gifted pupils of secondary schools in the Czech Republic. In the school-year 2010/2011 there was assigned for pupils on special seminarium (maths camp) the following problem: Problem 1. Let a, b, c be pairwisely different positive real numbers. Prove that the quadratic equation (a+b+c)x~2+2((a/b)+(b/c)+(c/a))x+((1/a)+(1/b)+(1/c))=0 with unknown x has two different real roots.
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