VMO 1983

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

(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.

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.

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

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?

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.