6th Mexican 1992


A1. The tetrahedron OPQR has the ∠POQ = ∠POR = ∠QOR = 90^o. X, Y, Z are the midpoints of PQ, QR and RP. Show that the four faces of the tetrahedron OXYZ have equal area.
A2. Given a prime number p, how many 4-tuples (a, b, c, d) of positive integers with 0 < a, b, c, d < p-1 satisfy ad = bc mod p?
A3. Given 7 points inside or on a regular hexagon, show that three of them form a triangle with area ≤ 1/6 the area of the hexagon.
B1. Show that 1 + 11¹¹ + 111¹¹¹ + 1111¹¹¹¹ + ... + 1111111111^1111111111 is divisible by 100.
B2. x, y, z are positive reals with sum 3. Show that 6 < √(2x+3) + √(2y+3) + √(2z+3) < 3√5.
B3. ABCD is a rectangle. I is the midpoint of CD. BI meets AC at M. Show that the line DM passes through the midpoint of BC. E is a point outside the rectangle such that AE = BE and ∠AEB = 90^o. If BE = BC = x, show that EM bisects ∠AMB. Find the area of AEBM in terms of x.

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