1st Vietnam 1962 problems

1.  Prove that 1/(1/a + 1/b) + 1/(1/c + 1/d) ≤ 1/(1/(a+c) + 1/(b+d) ) for positive reals a, b, c, d.
2.  f(x) = (1 + x)(2 + x2)1/2(3 + x3)1/3. Find f '(-1).
3.  ABCD is a tetrahedron. A' is the foot of the perpendicular from A to the opposite face, and B' is the foot of the perpendicular from B to the opposite face. Show that AA' and BB' intersect iff AB is perpendicular to CD. Do they intersect if AC = AD = BC = BD?
4.  The tetrahedron ABCD has BCD equilateral and AB = AC = AD. The height is h and the angle between ABC and BCD is α. The point X is taken on AB such that the plane XCD is perpendicular to AB. Find the volume of the tetrahedron XBCD.
5.  Solve the equation sin6x + cos6x = 1/4.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

Vietnam home
© John Scholes
7 July 2002