21st Vietnamese Mathematical Olympiad 1983 Problems

21st Vietnamese Mathematical Olympiad 1983 Problems

 A1.  For which positive integers m, n with n > 1 does 2ⁿ - 1 divides 2^m + 1?

A2.  (1) Show that (sin x + cos x) √2 ≥ 2 sin(2x)^1/4 for all 0 ≤ x ≤ π/2.

(2) Find all x such that 0 < x < π and 1 + 2 cot(2x)/cot x ≥ tan(2x)/tan x.

A3.  P is a variable point inside the triangle ABC. D, E, F are the feet of the perpendiculars from P to the sides of the triangles. FInd the locus of P such that the area of DEF is constant. 

B1.  For which n can we find n different odd positive integers such that the sum of their reciprocals is 1? 

B2.  Let s_n = 1/((2n-1)2n) + 2/((2n-3)(2n-1)) + 3/((2n-5)(2n-2)) + 4/((2n-7)(2n-3) + ... + n/(1(n+1)) and t_n = 1/1 + 1/2 + 1/3 + ... + 1/n. Which is larger? 

B3.  ABCD is a tetrahedron with AB = CD. A variable plane intersects the tetrahedron in a quadrilateral. Find the positions of the plane which minimise the perimeter of the quadrilateral. Find the locus of the centroid for those quadrilaterals with minimum perimeter.