1st Vietnam Mathematical Olympiad 1962 Problems

1st Vietnam Mathematical Olympiad 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.

Solutions

1. A straightforward, if inelegant, approach is to multiply out and expand everything. All terms cancel except four and we are left with 2abcd ≤ a²d² + b²c², which is obviously true since (ad - bc)² ≥ 0. 

2. Differentiating gives f '(x) = (2 + x²)^1/2(3 + x³)^1/3 + terms with factor (1 + x). Hence f '(-1) =3^1/22^1/3. 

3. Let the ray AB' meet CD at X and the ray BA' meet CD at Y. If AB' and A'B intersect, then X = Y. Let L be the line through A' parallel to CD. Then L is perpendicular to AA'. Hence CD is perpendicular to AA'. Similarly, let L' be the line through B' parallel to CD. Then L' is perpendicular to BB', and hence CD is perpendicular to BB'. So CD is perpendicular to two non-parallel lines in the plane ABX. Hence it is perpendicular to all lines in the plane ABX and, in particular, to AB.

Suppose conversely that AB is perpendicular to CD. Consider the plane ABY. CD is perpendicular to AB and to AA', so CD is perpendicular to the plane. Similarly CD is perpendicular to the plane ABX. But it can only be perpendicular to a single plane through AB. Hence X = Y and so AA' and BB' belong to the same plane and therefore meet.

4. Put a = sin²x, b = cos²x. Then a and b are non-negative with sum 1, so we may put a = 1/2 + h, b = 1/2 - h. Then a³ + b³ = 1/4 + 3h² ≥ 1/4 with equality iff h = 0. Hence x is a solution of the equation given iff sin²x = cos²x = 1/2 or x is an odd multiple of π/4. 

Source: Nguyễn Thị Lan Phương, http://321math.blogspot.com