MathDB

3

Part of 1984 Vietnam National Olympiad

Problems(2)

Finding points and loci for a given square...

Source: Vietnam MO 1984 P3

3/20/2011
A square ABCDABCD of side length 2a2a is given on a plane Π\Pi. Let SS be a point on the ray AxAx perpendicular to Π\Pi such that AS=2a.AS = 2a. (a)(a) Let M∈BCM \in BC and N∈CDN \in CD be two variable points. ii. Find the positions of M,NM,N such that BM+DN≥32BM + DN \ge \frac{3}{2}, planes SAMSAM and SMNSMN are perpendicular and BM⋅DNBM \cdot DN is minimum. iiii. Find MM and NN such that ∠MAN=45∘\angle MAN = 45^{\circ} and the volume of SAMNSAMN attains an extremum value. Find these values. (b)(b) Let QQ be a point such that ∠AQB=∠AQD=90∘\angle AQB = \angle AQD = 90^{\circ}. The line DQDQ intersects the plane π\pi through ABAB perpendicular to Π\Pi at Q′Q'. ii. Find the locus of Q′Q'. iiii. Let KK be the locus of points QQ and let CQCQ meet KK again at RR. Let DRDR meets Π\Pi at R′R'. Prove that sin2∠Q′DB+sin2∠R′DBsin^2 \angle Q'DB + sin^2 \angle R'DB is independent of QQ.
geometry unsolvedgeometry
Showing trigonometric relation and angle relation.

Source: Vietnam MO 1984 P6

3/20/2011
Consider a trihedral angle SxyzSxyz with \angle xSz = \alpha , \angle xSy = \beta and ∠ySz=γ\angle ySz =\gamma. Let A,B,CA,B,C denote the dihedral angles at edges y,z,xy, z, x respectively. (a)(a) Prove that sin⁡αsin⁡A=sin⁡βsin⁡B=sin⁡γsin⁡C\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C} (b)(b) Show that α+β=180∘\alpha + \beta = 180^{\circ} if and only if ∠A+∠B=180∘.\angle A + \angle B = 180^{\circ}. (c)(c) Assume that \alpha=\beta =\gamma = 90^{\circ}. Let O∈SzO \in Sz be a fixed point such that SO=aSO = a and let M,NM,N be variable points on x,yx, y respectively. Prove that ∠SOM+∠SON+∠MON\angle SOM +\angle SON +\angle MON is constant and find the locus of the incenter of OSMNOSMN.
trigonometrygeometryincentergeometry unsolved