11.2

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

cylindrical glass with water (II Soros Olympiad 1995-96 R1 11.2)

Source:

A cylindrical glass filled to the brim with water stands on a horizontal plane. The height of the glass is

2

times the diameter of the base. At what angle must the glass be tilted from the vertical so that exactly

1/3

of the water it contains pours out?

geometry3D geometry

arcsin (sinx) + arccos (cosx)=0 (II Soros Olympiad 1995-96 R2 11.2)

Source:

Solve the equation

arc \sin (\sin x) + arc \cos (\cos x)=0

algebratrigonometry

tetrahedron with heights 1,2,3,6 (II Soros Olympiad 1997-98 R3 11.2)

Source:

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers

1

2

3

6

geometry3D geometrytetrahedron