3rd Balkan 1986


A1. A line through the incenter of a triangle meets the circumcircle and incircle in the points A, B, C, D (in that order). Show that AB·CD ≥ BC²/4. When do you have equality?
A2. A point is chosen on each edge of a tetrahedron so that the product of the distances from the point to each end of the edge is the same for each of the 6 points. Show that the 6 points lie on a sphere.
A3. The integers r, s are non-zero and k is a positive real. The sequence a_n is defined by a₁ = r, a₂ = s, a_n+2 = (a_n+1² + k)/a_n. Show that all terms of the sequence are integers iff (r² + s² + k)/(rs) is an integer.
A4. A point P lies inside the triangle ABC and the triangles PAB, PBC, PCA all have the same area and the same perimeter. Show that the triangle is equilateral. If P lies outside the triangle, show that the triangle is right-angled.

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