6th USAMO 1977


1. For which positive integers a, b does (x^a + ... + x + 1) divide (x^ab + x^ab-b + ... + x^2b + x^b + 1)?
2. The triangles ABC and DEF have AD, BE and CF parallel. Show that [AEF] + [DBF] + [DEC] + [DBC] + [AEC] + [ABF] = 3 [ABC] + 3 [DEF], where [XYZ] denotes the signed area of the triangle XYZ. Thus [XYZ] is + area XYZ if the order X, Y, Z is anti-clockwise and - area XYZ if the order X, Y, Z is clockwise. So, in particular, [XYZ] = [YZX] = -[YXZ].
3. Prove that the product of the two real roots of x⁴ + x³ - 1 = 0 is a root of x⁶ + x⁴ + x³ - x² - 1 = 0.
4. ABCD is a tetrahedron. The midpoint of AB is M and the midpoint of CD is N. Show that MN is perpendicular to AB and CD iff AC = BD and AD = BC.
5. The positive reals v, w, x, y, z satisfy 0 < h ≤ v, w, x, y, z ≤ k. Show that (v + w + x + y + z)(1/v + 1/w + 1/x + 1/y + 1/z) ≤ 25 + 6( √(h/k) - √(k/h) )². When do we have equality?

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