MathDB

1

Part of 1992 Vietnam National Olympiad

Problems(2)

tetrahedron and its area surfaces

Source: 30-th Vietnamese Mathematical Olympiad 1992

2/17/2007
Let ABCDABCD be a tetrahedron satisfying i)ACD^+BCD^=1800\widehat{ACD}+\widehat{BCD}=180^{0}, and ii)BAC^+CAD^+DAB^=ABC^+CBD^+DBA^=1800\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}. Find value of [ABC]+[BCD]+[CDA]+[DAB][ABC]+[BCD]+[CDA]+[DAB] if we know AC+CB=kAC+CB=k and ACB^=α\widehat{ACB}=\alpha.
geometry3D geometrytetrahedrongeometry unsolved
root of polynomial

Source: 30-th Vietnamese Mathematical Olympiad 1992

2/17/2007
Let 9<n1<n2<…<ns<1992 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992 be positive integers and P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}. Prove that if x0 x_{0} is real root of P(x) P(x) then x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}.
algebrapolynomialalgebra unsolved