6th Junior Balkan 2002


1. The triangle ABC has CA = CB. P is a point on the circumcircle between A and B (and on the opposite side of the line AB to C). D is the foot of the perpendicular from C to PB. Show that PA + PB = 2·PD.
2. The circles center O₁ and O₂ meet at A and B with the centers on opposite sides of AB. The lines BO₁ and BO₂ meet their respective circles again at B₁ and B₂. M is the midpoint of B₁B₂. M₁, M₂ are points on the circles center O₁ and O₂ respectively such that angle AO₁M₁ = angle AO₂M₂, and B₁ lies on the minor arc AM₁ and B lies on the minor arc AM₂. Show that angle MM₁B = angle MM₂B.
3. Find all positive integers which have exactly 16 positive divisors 1 = d₁ < d₂ < ... < d₁₆ such that the divisor d_k, where k = d₅, equals (d₂ + d₄) d₆.
4. Show that 2/( b(a+b) ) + 2/( c(b+c) ) + 2/( a(c+a) ) ≥ 27/(a+b+c)² for positive reals a, b, c.

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