3rd APMO 1991


A1. ABC is a triangle. G is the centroid. The line parallel to BC through G meets AB at B' and AC at C'. Let A'' be the midpoint of BC, C'' the intersection of B'C and BG, and B'' the intersection of C'B and CG. Prove that A''B''C'' is similar to ABC.
A2. There are 997 points in the plane. Show that they have at least 1991 distinct midpoints. Is it possible to have exactly 1991 midpoints?
A3. x_i and y_i are positive reals with ∑₁ⁿ x_i = ∑₁ⁿ y_i. Show that ∑₁ⁿ x_i²/(x_i + y_i) ≥ (∑₁ⁿ x_i)/2.
A4. A sequence of values in the range 0, 1, 2, ... , k-1 is defined as follows: a₁ = 1, a_n = a_n-1 + n (mod k). For which k does the sequence assume all k possible values?
A5. Circles C and C' both touch the line AB at B. Show how to construct all possible circles which touch C and C' and pass through A.

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