MathDB

3

Part of 1985 IberoAmerican

Problems(2)

1st ibmo - colombia 1985./q3

Source: Spanish Communities

4/6/2006
Find all the roots r1 r_{1}, r2 r_{2}, r3 r_{3} y r4 r_{4} of the equation 4x^{4}\minus{}ax^{3}\plus{}bx^{2}\minus{}cx\plus{}5 \equal{} 0, knowing that they are real, positive and that: \frac{r_{1}}{2}\plus{}\frac{r_{2}}{4}\plus{}\frac{r_{3}}{5}\plus{}\frac{r_{4}}{8}\equal{} 1.
ColumbiaVietaalgebra unsolvedalgebra
1st ibmo - colombia 1985./q6

Source: Spanish Communities

4/6/2006
Given an acute triangle ABCABC, let DD, EE and FF be points in the lines BCBC, ACAC and ABAB respectively. If the lines ADAD, BEBE and CFCF pass through OO the centre of the circumcircle of the triangle ABCABC, whose radius is RR, show that: \frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}
Columbiageometrycircumcircletrigonometrygeometry unsolved