1st Vietnam 1962


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.

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